In this chapter we will be learning about powers and radicals. In this chapter we will be learning about powers and radicals. In this lesson we will summarize the laws of exponents. In this lesson we will summarize the laws of exponents. So, So here in this lesson we're going to look at the rules that govern how we use exponents.
here in this lesson, we're going to look at the rules that govern how we use exponents. Now, Now some of these rules are going to be familiar to you. some of these rules are going to be familiar to you.
Some of them are going to be a little bit new. Some of them are going to be a little bit new. So, So let's just take a quick look here.
let's just take a quick look here. In fact, In fact, let's maybe zoom in a little bit here. let's maybe zoom in a little bit here.
So first of all, So first of all we're going to start with the multiplication law. we're going to start with the multiplication law. If you've got two powers multiplied together and they have the same base, If you've got two powers multiplied together and they have the same base, then you simply add the exponents together. then you simply add the exponents together. Actually that makes perfect sense here.
Actually, that makes perfect sense here. Let's say we had x squared times x cubed. Let's say we had x squared times x cubed.
Well let's think about what that is. Well, let's think about what that is. That's x times x multiplied by x times x times x. That's x times x multiplied by x times x times x. Remember what the exponent does.
Remember what the exponent does. The exponent tells you how many times that that base there is a factor. The exponent tells you how many times that that base there is a factor.
Well, Well, when you put those together, when you put those together, you've got five factors here. you've got five factors here, so that's x to the fifth. So that's x to the fifth. That's really just the sum of those two.
That's really just the sum of those two. Now, Now, it's really important that you understand that this is how this works. it's really important that you understand that this is how this works.
This is how this always works, This is how this always works, okay? okay? Because there have been times when I've seen people that want to do this here, Because there have been times when I've seen people that want to do this here, this here, where you've got, where you've got, let's say, let's say, two squared times two cubed. 2 squared times 2 cube. Let's kind of use the same pattern here.
Let's kind of use the same pattern here. Now, Now, at this point here, At this point here, it should be clear that the answer here is 2 to the it should be clear that the answer here is 2 to the 5th. However, 5th.
However, I've watched people do this and write, I've watched people do this and write, well, well, that's got to be 4 to the 5th because you multiply the bases together. that's got to be 4 to the 5th because you multiply the bases together. No, No, you don't.
you don't. No, No, you don't. you don't. What you're doing here is you're multiplying the powers, What you're doing here is you're multiplying the powers, the entire thing. the entire thing.
You multiply the entire thing. You multiply the entire thing. And the way we do that is to leave the base the same and then to add the exponents. And the way we do that is to leave the base the same and then to add the exponents. Okay, that's how you do that.
That's how you do that. Don't think of it as multiplying the bases together, Don't think of it as multiplying the bases together, because you're not. because you're not.
You're multiplying the powers together, You're multiplying the powers together, and we do that by adding the exponents. and we do that by adding the exponents. Now, Now, obviously, obviously, the division law, the division law, the next one here, the next one here, is the opposite in terms of operation here.
is the opposite in terms of operation here. So when we've got two powers and we're dividing them, So when we've got two powers and we're dividing them, we simply subtract the exponents. we simply subtract the exponents. And it's for a very, And it's for a very, very similar reason. very similar reason.
If you've got, If you've got, let's say, let's say, x cubed over x squared, x cubed over x squared, That is x multiplied by x multiplied by x over x multiplied by x. that is x multiplied by x multiplied by x over x multiplied by x. And then we can take the factors here and group them together in these little pairs here. And then we can take the factors here and group them together in these little pairs here.
Because x divided by x is 1, Because x divided by x is 1, x divided by x is 1. x divided by x is 1. So because we have an xs on the top, So because we have an xs on the top, that's all we're left with. that's all we're left with. So, So yeah, yeah, x cubed over x squared would simply be x to the 1 left over.
x cubed over x squared would simply be x to the 1 left over. And there we go. There we go.
Next is the power law. Next is the power law. If we've got a power of a power, If we've got a power of a power.
and then what we just simply do here is we multiply the exponents together. And then what we simply do here is we multiply the exponents together. Now, Now... This is a rule here that causes a lot of problems later on with the math that you're going to be doing here.
this is a rule here that causes a lot of problems later on with the math that you're going to be doing. here just to explain what's going on here there's a couple things you need to know about this so if you've got a power of a power here you simply multiply the exponents together okay and again it's easy enough to explain why that works let's go over here let's say you have for example x cubed squared Just to explain what's going on here, there's a couple things you need to know about this. So if you've got a power of a power here, you simply multiply the exponents together.
And again, it's easy enough to explain why that works. Let's go over here. Let's say you have, for example, x cubed squared. Well, well what that square means is you've got x cubed multiplied by x cubed okay and then we would just apply the the rules here if you you would then add those two exponents together, what that square means is you've got x cubed multiplied by x cubed. Okay, and then we would just apply the rules here.
If you would then add those two exponents together, you get x to the sixth. you get x to the sixth. But just think about that. But just think about that. I have two I have two threes here.
3s here. Well, Well, that's multiplication, that's multiplication, right? right? When I'm adding the two 3s, When I'm adding the two threes, that's just 3 times 2. that's just three times two. And that's why that rule works.
And that's why that rule works. Now here's another aspect of it. Now, here's another aspect of it. If the part of the base here, If the part of the base here, if the base consists of multiple factors, if the base consists of multiple factors, then what happens is that exponent gets distributed to every factor in the base.
then what happens is that exponent gets distributed to every factor in the base. But I have to say this again. but I have to say this again, it's if your base consists of factors, It's if your base consists of factors, many factors.
many factors. Okay. Here's the problem that people are going to have later on here. Here's the problem that people are going to have later on here. Let's say that I've got X plus Y to some value A, Let's say that I've got x plus y.
to some value a, let's say. let's say. This does not equal, This does not equal, oops, oops, sorry.
sorry. This does not equal x to the a plus y to the a. This does not equal X to the A plus Y to the A.
There's a very, There's a very, very distinct difference between these two expressions. very distinct difference between these two expressions. And the difference here is that this is a monomial in here.
The difference here is that this is a monomial in here. This is a binomial in here. This is a binomial in here. These are multiplied together, These are multiplied together, so each of these here, so each of these here, the 2, the 2, the m, the m, and the n, and the n, are all factors of that monomial.
are all factors of that monomial. x and y are not factors here. x and y are not factors here.
These are what we would call addends. These are what we would call addends. You cannot distribute that over.
You cannot distribute that over. The rule is very different when you've got a binomial or larger as the basis. The rule is very different when you've got a binomial or larger as the basis.
that uh that power here please do not confuse these two rules oh please don't confuse them okay this people doing this right here causes the loss of so many potential marks uh in few that power here. Please do not confuse these two rules. Oh please don't confuse them.
Okay, this people doing this right here causes the loss of so many potential marks in future exams. future exams this problem right here. This problem right here.
You can only distribute over factors. You can only distribute over factors. Okay, And then the same thing happens over here if you've got a division, and then the same thing happens over here.
If you've got a division, sorry a quotient here, sorry a quotient here, you can distribute the you can distribute the The power here to both the numerator and the denominator, power here to both the numerator and the denominator. that's the rule for that. That's the rule for that.
Okay, Okay, now another one here. now another one here. This one here you should be totally familiar with. This one here you should be totally familiar with.
x to the 0 is equal to 1. X to the 0 is equal to 1. Now you might not understand why that occurs. Now you might not understand why that occurs. Actually the pattern is quite straightforward.
Actually, the pattern is quite straightforward. I'll just give you an example here. I'll just give you an example here.
Let's say you've got 2 cubed. Let's say you've got 2 cubed. You know that that's 8. You know that that's 8. 2 squared is 4. 2 squared is 4. All I did was I subtracted 1 from the exponent, All I did was I subtracted 1 from the exponent and then I took this value here and divided by 2. and then I took this value here and divided by 2. Let's do this.
Let's do that again. do that again. So 2 to the 1 is 2. So 2 to the 1 is 2. I subtracted 1 from the exponent and divided by 2 on the other side. I subtracted 1 from the exponent and divided by 2 on the other side. And now I'm just going to follow this pattern.
And now I'm just going to follow this pattern. If I subtract 1 from the exponent, If I subtract 1 from the exponent, I get 2 to the 0. I get 2 to the 0. And if I divide by 2, And if I divide by 2, I'm going to get 1. I'm going to get 1. That has to be true. That has to be true. It has to be true. It has to be true.
So it has to be the case that x to the 0 is equal to 1. So it has to be the case that x to the 0 is equal to 1. Now, now I know, I know I know when people look at that and they go oh it shouldn't I know when people look at that, they're like, oh, shouldn't that be 0? that be 0, but it's not x multiplied by 0, But it's not x multiplied by 0. it's x to the exponent 0. It's x to the exponent 0. So the rule is a little bit different. So the rule is a little bit different. Please do not misinterpret that and see that as a multiplication.
Please do not misinterpret that and see that as a multiplication. Now, Now, here we go. here we go. Negative exponent laws.
Negative exponent laws. Okay, Okay, if we take this little pattern here, if we take this little pattern here, right here, right here, one step further, one step further, the very next thing I would do here is to subtract 1 from that exponent I would get. the very next thing I would do here is to subtract 1 from that exponent. I would get 2 to the negative 1. 2 to the negative 1. And if I was to divide by 2 right here, And if I was to divide by 2 right here, I would get 1 over 2. I would get 1 over 2. So now notice what happens here. So now notice what happens here.
Notice that that negative exponent is equal to a reciprocal. Notice that that negative exponent is equal to a reciprocal. Okay, Okay, on the right-hand side here, on the right-hand side.
side here. it takes that base and reciprocates it. It takes that base and reciprocates it. If I was to go one step further here, If I was to go one step further here, this would become this would become 2 to the negative 2. 2 to the negative 2. And then what would happen is I would divide by 2 again, And then what would happen is I would divide by 2 again, and this would become 1 over and this would become 1 over 4, or 1 over 2 squared, 4, or 1 over 2 squared, if you want to think of it like that.
if you want to think of it like that. But notice what that negative in the exponent does. But notice what that negative in the exponent does.
Okay, It moves this expression here that's really in the numerator, it moves this expression here that's really in the numerator. this is like 2 to the negative 2 over 1, like 2 to the negative 2 over 1, and pushes it into the denominator. and pushes it into the denominator. And then it becomes a positive exponent.
And then it becomes a positive exposure. exponent, okay? Okay, That's what the negative exponent does.
that's what the negative exponent does. It doesn't change this thing into a negative value. It doesn't change this thing into a negative value, Because you got to understand this, because you've got to understand this, again, again, where people get confused by this is assuming that this is multiplication, where people get confused by this is assuming that this is multiplication, but it's not multiplication. but it's not multiplication, Okay, okay? it's a shorthand for referring to a bunch of multiplication, It's a shorthand for referring to a bunch of multiplication, but this is not multiplication in and of itself.
but this is not multiplication in and of itself, Okay, okay? so this negative does not make this whole term negative. So this negative does not make this whole term negative. In fact, In fact, it causes a reciprocal.
it causes a reciprocal. Or, Or, likewise, likewise, if you've got 1 over b to the negative 1, if you've got 1 over b to the negative 1, again, again, that negative 1 there is going to cause a reciprocal. that negative 1 there is going to cause a reciprocal. It's going to cause a thing to pop up.
It's going to cause a thing to pop up. Now, Now, finally here, finally here, we're going to take a look at fractional exponents. we're going to take a look at fractional exponents. We've already spent a little bit of time looking at this. We've already spent a little bit of time looking at this.
If you've got x to the 1 over m, If you've got x to the 1 over m, Yeah. We already know that that m becomes that root index, we already know that that m becomes that root index, that this is the same as a radical. that this is the same as a radical.
And if you've got x to the m over n, And if you've got x to the m over n, then it's the nth root of that power of x there, then it's the nth root of that power of x there, x to the m. x to the m. Or that exponent there can happen either inside the radical or outside the radical.
Or that exponent there can happen either inside the radical or outside the radical. Either one of those is totally acceptable. Either one of those is...
is totally acceptable. They're equivalent to each other. They're equivalent to each other. You will find, You will find, though, though, that this is often the easiest to work with, that this is often the... The easiest to work with, I guess, I guess is the best to say it.
is the best to say it. They're both really the same, They're both really the same, doing the same thing, doing the same thing, but it's often easier to reduce the radical first and then apply the power than it is to do that in the other order. but it's often easier to reduce the radical first and then apply the power than it is to do that in the other order. The following are examples of the multiplication law. The following are examples of the multiplication law.
Okay, Okay, so for these little, so for this little section of questions here, this little section of questions here, we're just going to apply the multiplication law here. we're just going to apply the exponent, the multiplication law here. So, So, x cubed times x squared, x cubed times x squared, we're simply going to add the exponents together, we're simply going to add the exponents together, we'll get x to the fifth.
we'll get x to the fifth. 3 to the exponent A multiplied by 3 to the exponent B. 3 to the exponent a, multiplied by 3 to the exponent b.
Well, Well, I don't know what A and B are. I don't know what a and b are. I don't know what they are, I don't know what they are.
But I do know that this operation tells me to add the exponents. but I do know that this operation tells me to add the exponents, So I can't go any further than that. so I can't go any further than that. So I would simply say here that I have to add the exponents.
So I will simply say here that I have to add the exponents. That's what that's expressing. That's what that's expressing. S to the fifth multiplied by S. S to the fifth multiplied by S.
I have to remember that if I've got a power sitting there, I have to remember that if I've got a power sitting there and the exponent is not written, the exponent is not written, that that implies that that's one. that that implies that that's one. So this would be S to the sixth when I add those together. This would be s to the 6th when I add those together.
And then over here, And then over here, it doesn't really matter here that this is a negative exponent here. it doesn't really matter here that this is a negative exponent here. When I multiply these two powers together, When I multiply these two powers together, it's still going to just add the exponents together first.
I'm still going to just add the exponents together first. Okay, so 4, So 4, this is going to be, this is going to be x to the 4 plus negative 2, sorry, x to the 4 plus negative 2, which is just 4, which is just x squared. whoops, sorry, x squared. Now keep going here. Now, keep going here.
This means the same thing. This means the same thing. I'm going to have 2 to the negative 2 multiplied by 2 cubed.
I'm going to have 2 to the negative 2 multiplied by 2 cubed. Now to start off with, Now, to start off with, again I see this as multiplication, again, I see this as multiplication, so I'm just going to multiply these together. so I'm just going to multiply these together. So I'm going to add the exponents. So I'm going to add the exponents.
And negative 2 plus 3 will be just And negative 2 plus 3 will be just... 1, 1. So this will be 2 to the 1. so this will be 2 to the 1. And I don't have to write the 1 if it is just 1. I don't have to write the 1 if it is just 1. Here, Here, 3 cubed multiplied by 3 to the negative 3, 3 cubed multiplied by 3 to the negative 3, well, well that's going to be 3 to the power of 3 plus negative 3. that's going to be 3 to the power of 3 plus negative 3. Now, Now that's just going to be 3 to the 0, that's just going to be 3 to the 0, and we've already seen that that is equivalent to 1. and we've already seen that that is equivalent to 1. Okay, Okay, now here. now here. Here we go.
Here we go. When you multiply these things together, When you multiply these things together, bear in mind that everything in here is being multiplied together. bear in mind that everything in here is being multiplied together. Like, Like, even though we're writing this differently here, even though we're writing this differently here, between the two parentheses here, between the two parentheses here, that implies multiplication. that implies multiplication.
The fact that these guys are all bunched up together, The fact that these guys are all bunched up together, there's nothing written in between, there's nothing written in between, implies multiplication. implies multiplication. These are all...
all multiplied together. all multiplied together. I am free to multiply in whatever order I wish.
I am free to multiply in whatever order I wish. So here's what I'm going to do. So here's what I'm going to do.
I'm going to rewrite this. I'm going to rewrite this. This will be 2 multiplied by 3. This will be 2 multiplied by 3. Can I move the 3 up?
Can I move the 3 up? Absolutely. Absolutely.
Order doesn't matter with multiplication. Order doesn't matter with multiplication. Make that x multiplied by x to the negative 1. that x multiplied by x to the negative 1. Can I do that?
Can I do that? Yes, Yes, I can. I can.
Order doesn't matter with multiplication. Order doesn't matter with multiplication. y squared, y squared, y to the 0. y to the 0, Can I do that?
can I do that? Yes, Yes, because once again, because once again, order doesn't matter with multiplication. order doesn't matter with multiplication. I can write that in whatever order I wish. I can write that in whatever order I wish.
Now, Now, I can multiply 2 times 3. I can multiply 2 times 3. That gets me 6. That gets me 6. x multiplied by x to the negative 1 is going to be x to the 1 plus negative 1, x multiplied by x to the negative 1 is going to be x to the 1 plus negative 1, and I know what's going to happen there. and I know what's going to happen there. And then this is going to be y, And then this is going to be y, 2 plus 0. 2 plus 0. Now, Now, this is going to end up being this is going to end up being 6, this will be x to the 0, 6. This will be x to the 0, and then 2 plus 0 is squared there.
and then 2 plus 0 is squared there. So I'm almost done here. So I'm almost done here.
I'll just go off to the side here. I'll just go off to the side here. x to the 0 is just 1, x to the 0 is just 1, and 1 times 6 multiplied by y squared is simply going to be 6 multiplied by y squared. and 1 times 6 multiplied by y squared is simply going to be 6 multiplied by y squared. So that's as simple as I can make that expression.
So that's as simple as I can make that expression. Now I hope what I did there makes sense. Now I hope what I did there makes sense.
You should be able to get to this answer a little bit quicker than what I did there, You should be able to get to this answer a little bit quicker than what I did there. but again, But again, there tends to be a little bit of confusion with this because of the multiple variables together and how to treat that. there tends to be a little bit of confusion with this because of the multiple variables together and how to treat that.
But the only things that I can simplify here, The only things that I can simplify here, even though they're all multiplied together, even though they're all multiplied together, the only thing I can simplify here are the powers of the same basis or the coefficients. the only thing I can simplify here are the powers of the same basis or the coefficients. So I am multiplying 6 by y squared here, I am multiplying 6 by 1 y squared here, but I can't go any further than this because I don't know what y is, but I can't go any further than this because I don't know what y is, and so that's a dead end. and so that's a dead end.
Until I know what numeric value this takes on, Until I know what numeric value this takes on, I cannot multiply it by the 6. I cannot multiply it by the 6. All I can do is state that it is being multiplied by the 6. All I can do is state that it is being multiplied by the 6. That's it. That's it. The following are examples of the division law. The following are examples of the division law.
All right, All right, in this section of questions here, in this section of questions here, we're going to apply the division law here. we're going to apply the division law here. Very similar to what we did with the previous set of questions here.
Very similar to what we did with the previous set of questions here. In this case, In this case, we're going to do subtraction here when we're dealing with the exponents. we're going to do subtraction here when we're dealing with the exponents.
So x to the 5th divided by x squared is going to be x to the 5 minus 2, So x to the fifth divided by x squared is going to be x. x to the 5 minus 3, sorry, 5 minus 2, which is going to be x cubed. which is going to be x cubed.
Over here, Over here, 3 to the m divided by 3 to the n. 3 to the m divided by 3 to the n. Now, Now in this case, in this case, I don't know.
I don't know what the exponent is. what the exponents are. exponents are. So the best I can do is just identify that that's what I would have to do.
So the best I can do is just identify that that's what I would have to do. Once I know what those numbers are, Once I know what those numbers are, I would need to subtract them to complete this operation. I would need to subtract them to complete this operation.
Okay, Okay, over here, over here's another way of writing that, just another way of writing that, and you should be familiar with this. and you should be familiar with this. So this is going to be x.
So this is going to be x. Now, Now x in the numerator, x in the numerator, when I do it like this, when I do it like this, to the 12 minus 3, to the 12 minus 3, and then 12 minus 3 is 9. and 12 minus 3 is 9. Okay? Okay, But we consider this in the numerator.
but we consider this in the numerator. When I bring that up as a subtraction up here, When I bring that up as a subtraction up here I'm assuming I'm assuming that what I'm left with is going to be in the numerator here. assuming that what I'm left with is going to be in the numerator here. Here, Here, don't let the fact that the exponent in the denominator is negative, don't let the fact that the exponent in the denominator is negative, don't let that throw you off. don't let that throw you off.
It doesn't matter. It doesn't matter. Just move ahead the way you know you should.
Just move ahead the way you know you should. The division here means that I'm going to take the exponents and subtract them. The division here means that I'm going to take the exponents and subtract them.
It's just in this case I'm subtracting a negative. It's just in this case I'm subtracting a negative. And so remember what you do when you subtract a negative.
And so remember what you do when you subtract a negative. You're going to add that, You're going to add that, and so you'll get x to the 6th there. and so you'll get x to the 6th there.
Okay, Okay, same thing here. same thing here. So this is going to become 5 to the exponent of 3 minus negative 4 for an overall result of 5 to the 7. So this is going to become 5 to the exponent of 3 minus negative 4 for an overall result of 5 to the 7. Okay, Okay, now here, now here...
there's a little bit more going on, There's a little bit more going on, and so what we want to do is we want to deal with order of operations. and so what we want to do is we want to deal with order of operations. We want to follow order of operations as strictly as we can here. We want to follow order of operations as strictly as we can here.
The bar always implies a set of brackets around the numerator and the denominator. The bar always implies a set of brackets around the numerator and the denominator, So the very first thing I want to do is simplify the numerator and simplify the denominator. so the very first thing I want to do is simplify the numerator and simplify the denominator.
Now, Now in this case, in this case, the numerator, the numerator, there are no common bases there amongst those factors, there are no common bases there amongst those factors, so I'm done. so I'm done. But in the denominator, But in the denominator, I've got a 2 multiplied by a 1, I've got a 2 multiplied by a 1, so it's going to be 2. so it's going to be 2. m squared multiplied by m, m squared multiplied by m.
Well, well I'm going to add the exponents together, I'm going to add the exponents together, so that'll be 3, so that'll be 3, because remember if it's not written, because remember, if it's not written, it's a 1. it's a 1. I've got n cubed multiplied by n squared, I've got n cubed multiplied by n squared, so I'm going to add the exponents, so I'm going to add the exponents, that'll become n to the 5th. that'll become n to the 5th. And so now, And so now, I'm left with, I'm left with, here, here, well, well I've still got this 1 half, I've still got this 1 half, I've still got this 2 in the denominator, I've still got this 2 in the denominator, so I'm just going to leave it like that. so I'm just going to leave it like that. And then I'm going to have m to the fifth minus three, And then I'm going to have m cubed, sorry, m to the 5th, minus 3, so I'm going to subtract the exponents.
so I'm going to subtract the exponents, Oops, oops you can't see that, you can't see that. And then I'll have n to the negative one minus five. and then I'll have n to the negative 1 minus 5. And so putting that together I will end up with m squared n to the negative 6 over 2. And so putting that together, I will end up with m squared n to the negative sixth over two.
two. And what I would do with this, And what I would do with this, and I'm running out of room down here, and I'm running out of room down here, so I'm just going to go over to the side here. so I'm just going to go to the side here. What I would do with this typically, What I would do with this typically, and it's not absolutely necessary, and it's not absolutely necessary.
there really is nothing wrong with a negative in the exponent, There really is nothing wrong with a negative in the exponent. okay? Okay.
You are going to have, You are going to have, there will be lots of people that will tell you that it's better form to not. there will be lots of people that will It will tell you that it's better form to not have negative exponents. have negative exponents. That's more of a personal preference thing.
That's more of a personal preference thing. There's really no rule that says you have to do it like that. There's really no rule that says you have to do it like that.
But you're going to see most people are going to take that and write it with a positive exponent by simply moving it to the other side of the fraction. But you're going to see most people are going to take that and write it with a positive exponent by simply moving it to the other side of the fraction. So this will be m squared over 2, So this will be m squared over 2 and then that n to the sixth simply moves into the denominator.
and then that n to the 6th simply moves into the denominator. And there you go. And there you go. Now finally in this case right here, Now finally in this case right here, it's very similar to the question that we just did.
it's very similar to the question we just did. We're just going to deal with the pieces that can be put together. We're just going to deal with the pieces that can be put together. So for example, So for example the 15 and the 20, the 15 and the 20, there's a common factor of 5 there which would leave me with there's a common factor of 5 there, which would leave me with 3 over 4. 3 over 4. Now I can't do anything else with that.
Now I can't do anything else with that. that, that's good. That's good. When I do the subtraction here, When I do the subtraction here, this will be m to the 7 minus 9 in the numerator.
this will be m to the 7 minus 9 in the numerator. Oops. Oops. n to the negative 1 minus negative 8. n to the negative 1 minus negative 8, Again, again, in the numerator when I do it like this. in the numerator.
When when I do it like this. And so now this will be 3 quarters m to the negative 2. And so now this will be 3 quarters m to the negative 2, N, n, now negative 1 minus negative 8, now negative 1 minus negative 8, be careful with that. be careful with that, We already know that as soon as we start throwing negatives around, we already know that as soon as we start throwing negatives around, a lot of people are going to make mistakes here, a lot of people are going to make mistakes here. so that should just be a clue to you that when you start to see these negatives showing up that you really do need to be careful with the arithmetic here.
So that should just be a clue to you that when you start to see these negatives showing up, that you really do need to be careful with the arithmetic here. Negative 1 minus negative 8 is going to be positive 7. Negative 1 minus negative 8 is going to be positive 7. And now, And now, because there's going to be a bit of a push on you to write things with positive exponents, because there's going to be a bit of a push on you to write things with positive exponents, we're going to take that m to the negative 2, we're going to take that m to the negative 2. We're going to move it to the other side of the rational here. we're going to move it to the other side of the rational here.
Whoops. Whoops. And there we go.
And there we go. So 3n to the 7 over So 3n to the 7 over 4m squared. 4m squared.
The following are examples of the power law. The following are examples of the power law. All right, All right, in this section of questions, in this section of questions, we're going to look at the power law.
we're going to look at the power law. So we've got x cubed squared. So we've got x cubed squared.
Remember what that means. Remember what that means. When you've got a power of a power, When you've got a power of a power, we're going to multiply the exponents together, we're going to multiply the exponents together, so this will become so this will become 3 times 2, 3 times 2, or x to the 6th.
or x to the 6th. Okay, and don't forget how this works. And don't forget how this works. As long as you've got factors here, As long as you've got factors here, as long as these are multiplied together, as long as these are multiplied together, okay, that there's only one term in there, that there's only one term in there, I can take that exponent and I can distribute it to the two factors here. I can take that exponent and I can distribute it to the two factors here.
So this will become m to the fifth, So this will become m to the fifth, and this will become n to the, and this will become m, sorry, n to the, and then it's going to be a power of a power, and then it's going to be a power of a power, so I'm going to multiply the exponents. so I'm going to multiply the exponents. This will become m to the fifth, This will become m to the fifth, nothing else to do there, nothing else to do there, n to the twentieth. n to the twentieth.
Here, Here, same thing, same thing, because there's parentheses here, because there's parentheses here, that two can distribute to both factors here. that 2 can distribute to both factors here. So this will become 3 squared, So this will become 3 squared, and then this will be z to the 6 times 2. and then this will be z to the 6 times 2. And the reason it just becomes 3 squared here is because the number that's the exponent on the 3 is 1, And the reason it just becomes 3 squared here is because the number that's the exponent on the 3 is a 1, and 2 times and 2 times 1 is 2. 1 is 2. We're just going to do that.
We're just going to do that. Now, Now, it makes more sense to write that as 9, it makes more sense to write that as 9, and then this would be z to the 12. and then this would be z. to the 12th.
Here, Here, very similar thing. very similar thing, and it's just that in this case that exponents will distribute to all of the things that are being either multiplied or divided together. And it's just that in this case, that exponents will distribute to all of the things that are being either multiplied or divided together.
Again, Again, as long as there's no, as long as there's no, we're not dealing with binomials or larger in here, we're not dealing with binomials or larger in here. We're okay to distribute the exponent. we're okay to distribute the exponent. You can't do it if there's a binomial or a larger polynomial in there.
You can't do it if there's a binomial or a larger polynomial in there. But in this case, But in this case, we're fine. we're fine. So this will become, So this will become whoops, Oops, sorry, sorry, I don't know why I started writing a 5 there.
I don't know why I started writing a 5 there. This will become r to the 2 times 3. This will become r. to the 2 times 3. That will distribute to the 2 here, That will distribute to the 2 here, so 2 cubed.
so 2 cubed. And this will be t to the And this will be t to the 5 times 3. 5 times 3. And then we can simply do the little bit of arithmetic that we're being told to do here. And then we can simply do the little bit of arithmetic that we're being told to do here.
So r to the 6, So r to the 6, this will be 8. this will be 8. t to the 15. t to the 15. Now this question right here. Now, this question right here. Now in this case order of operations is our kind of our guiding rule here. Now, in this case, order of operations is kind of our guiding rule here. So the first thing we're going to do will be the brackets, So the first thing we're going to do will be the brackets, what's inside the brackets.
what's inside the brackets. But there's really nothing to do here. There's really nothing to do here. So I'm going to do the exponent prior to doing the multiplication. So I'm going to do the...
exponent prior to doing the multiplication. So this will become b to the 2 times 3 times b to the 4 times 4. So this will become b to the 2 times 3 times b to the 4 times 4. I'm going to do this first here. I'm going to do this first here.
And this is going to get me b to the This is going to get me b to the 6th multiplied by b to the 16. 6th multiplied by b to the 16. Okay, Okay, now I can move down and deal with the multiplication. now I can move down and deal with the multiplication. Now that I've dealt with the exponents and the brackets here, Now that I've dealt with the exponents and the brackets here, now because the bases are the same, now, because the bases are the same, it will become B. it will become b, And I will add those exponents together because that's what the rule is for multiplication. and I will add those exponents together, because that's what the rule is for multiplication.
So my final answer here will be b to the 22. So my final answer here will be b to the 22. Here this is really no different than a couple of the previous ones we've done because everything is being multiplied together here. Here this is really no different than a couple of the previous ones we've done because everything is being multiplied together here. I can distribute that 2 to each of the exponents, I can distribute that 2 to each of the exponents, so 3 squared, so 3 squared, x to the 6 multiplied by 2, x to the 6 multiplied by 2, y to the 5 multiplied by 2. y to the 5 multiplied by 2. That's going to be 9, That's going to be 9, x to the 12th, x to the 12th, y to the 10th. y to the 10th.
There we go. There we go. The following are examples of the zero law. Following our examples of the zero law, Alright, Alright, we're just going to look at a few questions here that talk about the zero law here, we're just going to look at a few questions here that talk about the zero law here, what happens when zero is in the exponent.
what happens when zero is in the exponent. And this should be pretty straightforward, And this should be pretty straightforward, I'm hoping. I'm hoping. There is, There is, however, however, an opportunity for people to make mistakes here because they're simply not 100% an opportunity for people to make mistakes here because they're simply not 100% clear on our notation.
clear on our notation. Notation is very, Notation is very, very important in mathematics. very important in mathematics. But here we go, But here we go, 5 to the exponent 0, 5 to the exponent 0, that is simply going to be 1. that is simply going to be 1. Negative 5 to the exponent 0. Negative 5 to the exponent 0. Now it's important to understand in this case what the parentheses are doing here.
Now it's important to understand in this case what the parentheses are. doing here. It's identifying what the base of that power is. It's identifying what the base of that power is.
The entire base is negative 5. The entire base is negative 5. Now, Now anytime we take a base to an exponent 0, anytime we take a base to an exponent 0, except in the case of a base, except in the case of 0 here, in the case of zero here, that's going to end up being just one. that's going to end up being just 1. Now, Now, compare that, compare that, though, though, to this question, to this question, and I'll leave this visible here. and I'll leave this visible here.
Okay, Okay? notice in this case here, Notice in this case here, what is the base of the power? what is the base of the power? What is the power in this expression? What is the power in this expression?
Well, Well, the answer is a to the 0 is the power. the answer is a to the 0 is the power, the base is a. The base is a.
So negative 2 is not the base of that power. So negative 2 is not the base of that power. If I had parentheses around this whole thing, If I had parentheses around this whole thing, it would have been part of the base.
it would be part of the base. But as it stands right now, But as it stands right now, no it's not. no it's not. What that negative 2 is is a coefficient of the power.
What that negative 2 is, is a coefficient of the power. So this is really going to end up being negative 2 multiplied by, So this is really going to end up being negative 2 multiplied by, well, well a to the 0 is simply 1. a to the 0 is simply 1. And so then our result here is simply going to be negative 2. And so then our result here is simply going to be negative 2. two. Okay. Okay, now if that's not 100% Now, if that's not a hundred percent clear, clear, let's maybe go back to this one.
let's maybe go back to this one. Let's just throw another question here. Let's just throw another question here.
If this had been written like this, If this had been written like this, negative five to the zero, negative 5 to the 0, maybe here's another case here. maybe here's another case here. What's the power here?
What's the power here? Well, Well, the power here is the power here is five to the zero, 5 to the 0, not negative 5 to the 0. not negative five to the zero. Okay, Okay. not negative 5 to the 0. Not negative five to the zero. When, when you read that like that, When you read that like that, when, when people read this as negative 5 to the 0, when people read this as negative five to the zero, you are reading, you are reading, you are applying the order of operations incorrectly.
you are applying the order of operations incorrectly. When you read this off here, When you read this off here, you must deal with the exponent prior to the multiplication by negative 1. you must deal with the exponent prior to the multiplication by negative 1. Okay, that's just an order of operations issue and people do this wrong. This is just an order of operations issue if people do this wrong.
This really should be understood to be negative 1 multiplied by 5 to the 0. This really should be understood to be negative 1 multiplied by 5 to the 0. Now, Now, you do the exponent first. you do the exponent first. That's what the order of operations tells us to do. That's what the order of operations tells us to do. 5 to 0 is 1. 5 to 0 is 1. And 1 multiplied by negative 1 is simply negative 1. and 1 multiplied by negative 1 is simply negative.
negative 1. So that negative survives because it's not part of the base. So that negative survives because it's not part of the base. Anyway, Anyway, I hope that's clear.
I hope that's clear. It just, You need some clarity on the notation, you need some clarity on the notation and really you need to be applying order of operations correctly. and really you need to be applying order of operations correctly. Let's take a look at one more here.
Let's take a look at one more here. Now in this particular case, Now in this particular case, the whole purpose of this questionnaire is just so that you understand that it doesn't really matter how complicated that base is. the whole purpose of this question here is just so that you understand that it doesn't really matter how complicated that base is. It makes no difference whatsoever. Makes no difference whatsoever.
The whole thing here, The whole thing here, The base here is this negative 2aq. the base here is this negative 2a cubed b squared, a cubed b squared, but the exponent is 0, but the exponent is 0, so the whole thing is just going to go to 1. so the whole thing is just going to go to 1. The following are examples of the negative exponents law. Examples of the negative exponents law.
Right. Right. Now we're going to take a look at a handful of questions here that basically push us to work with negative exponents. Now we're going to take a look at a handful of questions here that... basically push us to work with negative exponents.
Okay? Okay, so here we got two-thirds to the negative four here. So here, we got 2 thirds to the negative 4 here.
My advice to you is deal with the negative, My advice to you is deal with the negative, like apply the order of operations, like, apply the order of operations, but once you're looking at exponents, but once you're looking at exponents, deal with the negative first. deal with the negative first. Okay, Okay? think of this as like negative one multiplied by four.
Think of this as like negative 1 multiplied by 4. You can just do them one after another here. And you just do them one after another here. Now what that negative does, Now, what that negative does, maybe I should write it like this.
maybe I should write it like this. This might help here. This might help here.
To the negative one to the four. To the negative 1 to the 4. So they're multiplied together here. said their mother multiply it together here. You can see this. You can see that.
So you separate that out here. You can separate that out here. What that negative 1 does is causes a reciprocal in the base. What that negative 1 does is causes a reciprocal in the base.
Okay? Okay? It causes a reciprocal in the base here. It causes a reciprocal in the base here. Now, Now, at least that's one way of interpreting it.
at least that's one way of interpreting it. We're going to see in the next question here that there's another way of thinking about that. We're going to see in the next question here that there's another way of thinking about that.
Now you're going to apply the 4 to the numerator and denominator, Now you're going to apply the 4 to the numerator. numerator and denominator, so 3 to the 4 over 2 to the 4. so 3 to the 4 over 2 to the 4. Okay, Okay, and when you do that and evaluate that, and when you do that and evaluate that, you'll get 81 over we'll get 81 over 16. 16. Now let's go over here. Now let's go over here. Although we can still interpret it like this, Although we can still interpret it like this, you might think of this as m to the negative 1 cubed, you might think of this as m to the negative 1 cubed, m to the negative 2, m to the negative 2, sorry, sorry, negative 1 squared.
negative 1 squared. Okay, Okay, that's one way of dealing with this. that's one way of dealing with this. And then what you could do is you could flip these things around here. And then what you could do is you could flip these things around here.
And so you would make this, And so you would make this, for example, for example, that negative one is going to cause. that negative one is going to cause... Now, Now, maybe in this case, maybe in this case, because the base is not a fraction, because the base is not a fraction, another way to think about it might be that it moves it to... another way to think about it might be that it moves it to the other part of the rational here. to the other part of the rational here.
So this is going to move down. So this is going to move down. This 3 stays with it.
This three stays with it. Here, Here, this case that moves up, this case that moves up, 2 stays with it here. two stays with it here. That's one way that you could do this. That's one way that you could do this.
And then you could make this. And then you could make this... this would be to be the same as, The same as, sorry, sorry, I'll just do it down here. I'll just do it down here.
Now I can deal with the subtraction of the exponents. Now I can deal with the subtraction of the exponents. So m to the 2 minus 3, So m to the 2 minus 3, which ends up being negative 1. which ends up being negative 1. And oh, And oh, look at that.
look at that, I got a negative exponent there. I got a negative exponent there. That negative exponent's going to force me to the other side of the fraction.
That negative exponent's going to force me to the other side of the fraction. So I get that there. So I get that there. That's one way of doing this. That's one way of doing this.
Now I've probably made that a little bit more difficult than it needs to be. Now I've probably made that a little bit more difficult than it needs to be. Okay. Okay. Come back up to this.
Come back up to this notice. Notice that we've got two powers of m where the bases are the same. Notice that we've got two powers of m where the bases are the same. I think you'd be smarter in this particular case to simply do this.
I think you'd be smarter in this particular case to simply do this. Make this negative 3 minus negative 2. Make this negative 3 minus negative 2. Just apply, Just apply, because it's a division here, because it's a division here, apply the rule that gives you subtraction here. apply this.
It gives you subtraction here. Now negative 3 minus negative 2 is going to be negative Now negative 3 minus negative 2 is going to be negative 1. 1. Be careful when you're dealing with the negatives. Be careful when you're dealing with the negatives.
And now, And now, as we just said here, as we just said here, that negative is simply going to cause a reciprocal in the base. that negative is simply going to cause a reciprocal in the base. So instead of being m to the 1, so instead of being m to the 1. This will be 1 over m to the 1. this will be 1 over m.
over m to the 1. Over here, Over here, you might want to do the exact same thing. you might want to do the exact same thing. Notice that that negative 2 is a coefficient. Notice that that negative 2 is a coefficient.
Please don't include the negative 2 here. Please don't include the negative 2 here. One way that you could do this, One way that you could do this, for example, for example, would be to move this x cubed down.
would be to move this x cubed down to the denominator. of the denominator. You would think of this as negative 2 times x to the negative 1 cubed all over x squared.
You would think of this as negative 2 times x to the negative 1 cubed all over x squared. And so one thing that you could do immediately is to simply move that x cubed down to the denominator. And so one thing that you could do immediately is to simply move that x cubed down to the denominator. So you've got like x cubed, So you've got like x cubed, and then you've got x squared, cubed x squared down here and leave the negative 2 up top.
x squared down here, and leave the negative 2 up top. Now I've seen, Now I've seen this is a method that a lot of people choose to use here. this is a method that a lot of people choose to use here. The mistake is to move the 2 with that. The mistake is to move the 2 with that.
But the 2 isn't part of the base. But the 2 isn't part of the base, It's not part of the power at all. it's not part of the power at all, It's a coefficient.
it's a coefficient. And then in the denominator, And then in the denominator, what you would do here would be because you're multiplying two powers of x together, what you would do here would be... because you're multiplying two powers of x together, you would add the exponents, you would add the exponents.
and so this would become x to the fifth, And so this would become x to the fifth. That That is one way of doing it. okay?
That is one way of doing it. Now, Now, again, again, probably not the best way to do that, probably not the best way to do it. to do that, although you should be aware that that is something that you could do. although you should be aware that that is something that you could do.
You'd probably be smarter, You'd probably be smarter, though, though, because you've got division here, because you've got division here, to simply apply the rule for division. to simply apply the rule for division. So this would become negative three minus two.
So this will become negative 3 minus 2. So negative two x to the negative five. So negative 2x to the negative 5 Okay, and again, And again, that negative on that x is going to force a reciprocal to happen, that negative on that x is going to force a reciprocal to happen, but it doesn't apply to the negative 2. but it doesn't apply to the negative 2. The negative 2 stays there in the numerator, The negative 2 stays there in the numerator, and then that x to the 5th simply moves into the denominator. and then that x to the 5th simply moves into the denominator. With this last question in this section here, With this last question in this section here, we're going to apply an order of operations. we're going to apply an order of operations.
We're going to deal with what's inside here first. We're going to deal with what's inside here first. Now, Now, there's not a lot of simplification that can happen here because these two powers have different bases.
there's not a lot of simplification that can happen here because these two powers have different bases. But, It's going to be argued that it's simpler if I have the y to a positive exponent. it's going to be argued that it's simpler if I have the y to a positive exponent. That y to the negative 3, That y to the negative 3, that negative is going to force it up into the numerator, that negative is going to force it up into the negative 3. The numerator, so this will become x squared, so this will become x squared, y cubed, y cubed, cubed. cubed.
So now I've dealt with what's inside the parentheses. So now I've dealt with what's inside the parentheses. Now that 3 will distribute to both those factors inside here. Now that 3 will distribute to both those factors inside here, And again, and again, I can do this because it's multiplication. I can do this because it's multiplication.
If it was a binomial or larger in there, if it was a binomial or larger in there, I couldn't do what I'm about to do here. I couldn't do what I'm about to do here. And so this becomes x to the 2 times 3, So this becomes x to the 2 times 3, y to the 3 times 3. y to the 3 times 3. So this becomes x to the So this becomes x to the 6th, y to the 9th.
6th, y to the 9th. The following are examples of the rational exponents law. The following are examples of the rational exponents law.
Right, Right, now in this section of questions here, now in this section of questions here, this is a little bit of review because we've looked at rational exponents before, this is a little bit of review, because we've looked at rational exponents before, but just to go over this ground again here. but just to go over this ground again here. So I've got 25 to the 1 half, So I've got 25 to the 1 half.
okay? Okay, Remember that that denominator there forms my root index, remember that that denominator there forms my root index. so this is the square root of 25. So this is...
The square root of 25. Now, Now, I really encourage you to be very, I really encourage you to be very, very clear with your notation here. very clear with your notation here, Okay, okay? math, Math, a huge, a huge, a huge chunk of mathematics is communication.
a huge chunk of mathematics. is communication. It's one thing for you to understand something, It's one thing for you to understand something, but if you cannot communicate to somebody else that you understand it, but if you cannot communicate to somebody else that you understand it, for all intents and purposes, for all intents and purposes, you don't understand it. you don't understand it. And that sounds kind of harsher, That sounds kind of harsher, but really that's the case here.
but really that's the case here. You need to be able to communicate. You need to be able to communicate.
Notation is very, notation is very, very important in this process here. very important in this process here. That 2, That 2, if it's a square root, if it's a square root, either you don't write it, either you don't write it, or you make sure it's clearly within the root symbol there.
or you make sure it's clearly within the root symbol there. Okay, if that 2 ends up out front, If that 2 ends up out front, it's a coefficient. it's a coefficient. You have miscommunicated.
You have miscommunicated. It looks like you don't understand what you're doing. It looks like you don't understand what you're doing.
And so now this is the square root of 25, And so now this is the square root of 25, and we know that what that means is, and we know that what that means is, okay, and I don't have to write that square root anymore, and I don't have to write that square root anymore. We would write that 25 is 5 times 5, We would write that 25. is 5 times 5, and a square root takes two factors and spits out a single factor, and a square root takes two factors and spits out a single factor, a single representative there. a single representative there. So the answer is 5. So the answer is 5. Okay, Okay, now we'll do this a little bit quicker here. now we'll do this a little bit quicker here.
So 36 to the 3 halves, So 36 to the 3 halves. my root index is 2. My root index is 2. I don't need to write it. I don't need to write it. And the exponent here, And the exponent here, the power there is going to be, the power there is going to be, sorry, sorry, the exponent, the exponent, not the power. not the power.
The power is the whole thing. The power is the whole thing. The exponent there is going to be 3. The exponent there is going to be 3. Now, Now, I told you before that you're probably smarter to do it like this, I told you before that you're probably smarter to do it like this.
to do the exponent here later, the exponent here later. apply that after the radical. Apply that after the radical. 36, 36, I would break that down into its prime factors, I would break that down into its prime factors, which will be 2 times 2 times 3 times which will be 2 times 2 times 3 times 3. 3. I can bring out a... I can bring out a 2, A 2, I can bring out a 3, I can bring out a 3. Okay, because I've got a pair of 2s and a pair of 3s.
because I've got a pair of 2s and a pair of 3s. Now, Now, what that's going to do is it's going to give me a 6 outside here. what that's going to do is it's going to give me a 6 outside here.
Okay, and 6 cubed is going to be 216. And 6 cubed is going to be 216. Over here, Over here, there's not a whole lot I can do here because I don't know what the base is. there's not a whole lot I can do here because I don't know what the base is. So I'm not going to be able to give you a number here. So I'm not going to be able to give you a number here.
So the best that I can do here is interpret what that rational exponent is giving me, So the best that I could do here is interpret what that rational exponent is. is giving me, and that is now giving me a cube root, and that is now giving me a cube root, and in this case I need to write the root index to be clear that that is a cube root. and in this case I need to write the root index to be clear that that is a cube root. The base is going to be a here, The base is going to be a here, and I would probably put parentheses out there and put a 2 out there, and I would probably put parentheses out there and put a 2 out there, put that exponent outside. put that exponent outside.
It could go inside, It could go inside, it's typically easier if it's outside. it's typically easier if it's outside. All right, All right, now here, now here, this one, this one's got a lot going on here.
this one's got a lot going on here. We've got to be really careful about this one. We've got to be really careful about this one. First of all, First of all, let's identify the power and what's going on here.
let's identify the power and what's going on here. Where's the power? Where's the power? Well, Well, the power is right here.
the power is right here. What's the base of the power? What's the base of the power?
This is important. This is important. Well, Well, it's just x.
It's without parentheses here. it's just x. Without parentheses here, The base of the power is just x. the base of the power is just x. Negative 3 is a coefficient.
Negative 3 is a coefficient. Okay? Okay?
So this really should be written as negative 3, So this really should be written as negative 3x to the negative 1 fifth. x, Okay, to the negative 1 5th. Now, Now, let's think of this more like this.
let's think of this. More like this. Base is x, Base is x, and this is to the negative 1 to the 1 and this is to the negative 1 to the 1 5th. So let's just deal with this in order here.
5th. So let's just deal with this in order here. The negative is going to cause a reciprocal to happen, The negative...
is going to cause a reciprocal to happen. So it's going to take that power, so it's going to take... Take that power, now just the power, now just the power, not the coefficient, not the coefficient, just the power. just the power, it's going to move it to the denominator.
It's going to move it to the denominator. So this becomes negative 3 over x to the 1 fifth. So this becomes negative 3 over x to the 1 fifth. And now because the purpose of this set of questions here is to deal specifically with the rational. And now because the purpose of this set of questions here is to deal specifically with the rational exponents, exponents, what I'm going to do here is I will rewrite this.
what I'm going to do here is I will rewrite this, And I'm not going to be able to do much with it because I don't know what x is. and I'm not going to be able to do much with it because I don't know what x is, But I can at least rewrite this in the right form and make that the fifth root of x. but I can at least rewrite this in the right form and make that the fifth root of x. That's what we're looking for. That's what we're looking for.
And we want to make sure that that five is written in the correct space here. And we want to make sure that that 5 is written in the correct space here. All right, All right.
in this question right here, And this question right here, in contrast to the previous one that we just did here, in contrast to the previous one we just did here, we're going to identify the power. we're going to identify the power. What's the power here?
What's the power here? Well, Well, the power is 4x cubed to the 1 half. the power is 4x cubed to the 1 half. The base, The base in this case is all of that.
in this case, is all of that. There is no coefficient. There is no coefficient.
I mean, coefficient. somebody can argue, Somebody can argue, well, well, wait a minute, wait a minute, wait a minute, yeah, yeah, it is. it is, it's going to end up being 2. It's going to end up being 2. And I know where you're going with that, And I know where you're going with that. But for right now, but for right now, just looking at the power, just looking at the power the way it's written right now, the way it's written right now, there's no coefficient. there's no coefficient. The base is The base is 4x cubed.
4x cubed. So to write this in radical form here, So to write this in radical form here, this is going to be the square root, this is going to be the square root, because it's one half, because it's 1 half, of 4x cubed. of 4x cubed.
Okay, Okay, now we should probably simplify that. now we should probably simplify that. This is the square root of 2 times 2 times x times x times x. This is the square root of 2 times 2 times x times x times x.
And I'm looking for pairs of factors, And I'm looking for pairs of factors, so I've got a pair of 2s and a pair of xs. so I've got a pair of 2s and a pair of xs. So So 2x, and there's still a root x that I couldn't take out. 2x, and there's still a root x that I couldn't take out. Okay, Okay, that x was left over.
that x was left over. And now finally this one right here, And now finally this one right here, a similar sort of question here. a similar sort of question here. I've got to look at this and identify what is the power here. I've got to look at this and identify what is the power here.
Well, Well, the whole thing is the power. the whole thing is the power. The base is x to the sixth y. The base is x to the sixth y.
And so what I'm going to do here is simply write this in radical notation. And so what I'm going to do here is simply write this in radical notation. So this is going to be the cube root of x to the sixth y.
So this is going to be the cube root of x to the sixth y. Now, Now, I'm going to be a little bit lazy here. I'm going to be a little bit lazy here. I'm not going to write out six x's here.
I'm not going to write out six x's here. But I do know that I'm looking for triplets. But I do know that I'm looking for triplets.
And if I've got six of them, And if I've got six of them, I'm going to have two groups of three x's, I'm going to have two groups of three x's, which means I can pull out an x squared. which means I can pull out an x squared. Now, Now, I've still got a cube root here because I've still got something underneath it. I've still got a cube root here because I've still got something underneath it. That y, That y, there was just a single y.
there was just a single y. There's not much I can do there. There's not much I can do there. So that's the answer to this thing in simplified form. So that's the answer to this thing in simplified form.
The following are just mixed examples of all of the previous laws. The following are just mixed examples of all of the previous laws. Okay, Okay, now to finish off here, now to finish off here, we're going to take a look at a group of questions here where we're not really specifically told what rule to apply. we're going to take a look at a group of questions here where we're not really specifically told what rule to apply. We're just told to simplify this.
We're just told to simplify this. Now, Now, simplifying in this case is really going to mean, simplifying in this case is really going to mean, for the most part here, for the most part here, basically... basically taking the exponent and making it the smallest possible number that we can. taking the exponent and making it the smallest possible number that we can. Okay?
Okay? Whether it's positive or negative, Whether it's positive or negative, we're trying to get it closer and closer to zero here. we're trying to get it closer and closer to zero here. Now, Now, notice that it says simplify each, notice that it says simplify each, leave all the answers to positive exponents. leave all the answers to positive exponents.
That is not necessarily part of simplifying. That is not necessarily part of simplifying. That's something we've got to state extra to that. That's something we've got to state extra to that. So just be aware that that's what's going on here.
So just be aware that that's what's going on here. So anyway, So anyway, here we go. here we go.
So b to the fifth multiplied by b divided by b squared. So b to the fifth multiplied by b divided by b squared. Now there are some shortcuts we can take here, Now there are some shortcuts we can take here, but because we're just introducing this topic right now, but because we're just introducing this topic right now, let's focus on the rules here that we've got to...
let's focus on the rules here that we've got to apply. Apply. First of all, First of all, the bar right there implies a set of brackets around the numerator and denominator.
the bar right there implies a set of brackets around the numerator and denominator. Order of operations tells me to deal with what's in brackets first. Order of operations tells me to deal with what's in... brackets first. So the first thing I'm going to do is this multiplication that shows up in the numerator.
So the first thing I'm going to do is this multiplication that shows up in the numerator. When I multiply two powers that have the same base, When I multiply two powers that have the same base, I'm going to add the exponents. I'm going to add the exponents. Okay, Okay, that's all I'm going to do.
that's all I'm going to do. Now this is going to end up being b to the sixth divided by b squared. Now this is going to end up being b to the 6th divided by b squared. Once that's done, Once that's done, division is going to become subtraction.
division is going to become subtraction. And so this is going to become b to the 4. And so this is going to become b to the 4. That's my final answer. That's my final answer.
Now, Now, I did that in several steps. I did that in several steps. I'm hoping that you'd be able to do this a little bit quicker, I'm hoping that you'd be able to do this a little bit quicker, that you would see what's going on here, that you would see what's going on here, that you would see, that you would see, for example, for example, this is going to be b to the this is going to be b to the 5 plus 1 minus 2, 5 plus 1 minus 2. okay?
Because you're going to do the multiplication first, Because you're going to do the multiplication first, and then you're going to do the subtraction here. and then you're going to do the subtraction here. You still get the same results. You still get the same results here.
You can do it a little bit quicker here. You can do it a little bit quicker here. The more comfortable you get, The more comfortable you get, everybody, everybody, the more comfortable you get with the rules.
the more comfortable you get with the rules the that we are applying here, that we are applying here, the faster you're going to get through this. the faster you're going to get through this. Okay?
The more comfortable you are with this, The more comfortable you are with this, the more happy you're going to be doing something like this. the more happy you're going to be doing something like this. Anyway, Anyway, let's keep going here. let's keep going here.
Same thing, Same thing, that line implies parentheses, that line implies parentheses, so I'm going to work with the power of the power first. so I'm going to work with the power of the power first. Okay?
I still would have worked with that anyway, I still would have worked with that anyway, because I'm going to deal with the exponents first. because I'm going to deal with the exponents first. This becomes y to the This becomes y to the 7th over y to the 2 times 3, 7th over y to the 2 times 3. which will be y to the 7 over y to the which will be Y to the 7th.
7 over y to the 6th. And now division is going to become subtraction, 6th, and now division is going to become subtraction, and 7 minus 6 is just 1, and 7 minus 6 is just 1. And if the exponent's 1, and if the exponent's 1, I don't need to write it. I don't need to write it. Over here, Over here, first thing we're going to do is deal with what's inside the exponents.
first thing we're going to do is deal with what's inside the exponents. Okay, Okay, well there's really no simplification that can occur within the... well there's really no simplification that can occur within the, Sorry, sorry, I said exponents here, I said exponents here, I meant brackets.
I meant brackets. Okay, Okay, 2a to the to the fourth, there's no simplification, 4th, there's no simplification there, there's no simplification there. so order of operations tells me that the next thing I'm going to deal with is the exponents. So, order of operations tells me that the next thing I'm going to deal with is the exponents.
Because these are, Because these are... Well, well this is, this is, in particular here, in particular here, this is just a monomial with two factors here. this is just a monomial with two factors here. I'm going to distribute the 3 through, I'm going to distribute the 3 through, so this will become 2 cubed, so this will become 2 cubed. multiplied by a to the 4 times 3. multiplied by a to the 4 times 3. Now this is all in brackets here.
Now, this is all in brackets here. And I'm going to do the same thing over here, And I'm going to do the same thing. same thing over here, and I might as well do it right now, and I might as well do it right now, because they are separated by a multiplication sign. because they are separated by a multiplication sign.
So I'm going to deal with exponents across the whole thing. So I'm going to deal with exponents across the whole thing. So this will be a to the 3 times 2. So this will be a to the 3 times 2. So right now I'm going to get 8, So right now I'm going to get 8, a to the 12, a to the 12, multiplied by a to the sixth.
multiplied by a to the 6th. Now I'm multiplying these together, Now, I'm multiplying these together, but I'm also multiplying these together. but I'm also multiplying these together.
Everything's being multiplied together here. Everything's being multiplied together here, So order doesn't really matter. so order doesn't really matter.
I can drop the parentheses here. I can drop the parentheses here. And when you multiply two powers together with the same base, And when you multiply two powers together with the same base, and by the way, and by the way, these are the only two that I can actually multiply together, these are the only two that I can actually multiply together, base stays the same, base stays the same, and I'm simply going to add the exponents together. and I'm simply going to add the exponents together. And so this will become 8 multiplied by a to the 18. And so this will become 8 multiplied by a to the 18th.
Okay, Okay, now, now again, again, I'm going to be really strict with this. I'm going to be really strict with this. The very first thing I'm going to do here is deal with what's inside the parentheses here. The very first thing I'm going to do here is deal with what's inside the parentheses here. Now, Now, I was told to simplify using positive exponents.
I was told to simplify using positive exponents. So now, So now there is a negative exponent here, there is a negative exponent here, so I'm going to deal with that first. so I'm going to deal with that first.
So that negative exponent is going to push that into the denominator, So that negative exponent is going to push that into the denominator, but just that. but just that, just that x. Just that x.
Okay? Okay, because when I look at that power here, Because when I look at that power here, the base of that power is just the x. the base of that power is going to be x.
power is just the x. The negative 2 is a coefficient. The negative 2 is a coefficient.
Okay, Unless there were parentheses around that in particular with that negative 1 as an exponent. unless there were parentheses around that in particular with that negative 1 as an exponent. No, No, just the x is the base here. just the x is the base here. Now, Now, what I'm going to do here, what I'm going to do here, because everything is being multiplied or divided here, because everything is being multiplied or divided here, that 3 can distribute to each of the exponents.
that 3 can distribute to each of the exponents. Now, Now, I'm going to treat this as negative 2 cubed. I'm going to treat this as negative 2 cubed.
So negative 2 cubed, so negative 2 cubed the whole thing here the whole thing here is going to be the base, Here's going to be the base because that 3 will be distributed to everything. because that 3 will be distributed to everything. So it will cover both the negative there and the value. So it'll cover both the negative there and the value. This will be y to the 4 times 3, This will be y to the 4 times 3 because I've got a power of a power.
because I've got a power of a power. And then x cubed. And then x cubed. The exponent on the x is 1, The exponent of the x is 1. just multiply it through. Just multiply it through.
And so negative 2 cubed is going to be negative 8. And so negative 2 cubed is going to be negative 8. This will be y to the 12 over x cubed. This will be y to the 12 over x cubed. Whoops, Whoops.
There we go. there we go. Okay.
Okay, Here we go. here we go. Now. Now, I'm going to multiply these together, I'm going to multiply these together, but first I'm going to do is look at what's inside the parentheses here.
but first I'm going to do is look at what's inside the parentheses here. There's really no simplification I can do, There's really no simplification I can do, other than the fact that I was told initially that I want to write these with positive exponents. other than the fact that I was told... initially that I want to write these with positive exponents. So I'm going to deal with this one first here.
So I'm going to deal with this one first here. So this will be negative So this will be negative 3m squared n, 3m squared n, negative negative 4m to the fourth over n squared. 4m to the fourth over n squared. All that happens is that negative moves it to the other side here.
All that happens is that negative moves it to the other side here. The 2 goes with it. The 2 goes with it.
Okay, He just moves to the other side of that rational. it just moves to the other side of that rational factor. that rational factor. Now Now I'm going to multiply.
I'm going to multiply. Now, Now bear in mind though that that n squared here is in the denominator and this n right here is in the numerator. bear in mind, though, that that n squared here is in the denominator, and this n right here is in the numerator. Now, Now you got to be careful when you multiply fractions together. you've got to be careful when you multiply fractions together.
This tends to be a point where This tends to be a point where I'm seeing a lot more people make mistakes here. I'm seeing a lot more people make mistakes here. This does not distribute to both the numerator and denominator. This does not distribute to both the numerator and denominator.
You simply multiply numerators together, You simply multiply numerators together, multiply denominators together. multiply denominators together. Okay, Please make sure you're doing that correctly. please make make sure you're doing that correctly. Negative 3 multiplied by negative 4. Negative 3 multiplied by negative 4. I can multiply the coefficients together.
I can multiply the coefficients together. They're both numbers. They're both numbers. I see what they are. I see what they are.
Negative 3 times negative 4 is positive 12. Negative 3 times negative 4 is positive 12. m squared multiplied by m to the fourth. M squared multiplied by M to the fourth. I'm going to add the exponents together.
I'm going to add the exponents together. n, n, well, well, there isn't an n in the numerator, there isn't an n in the numerator, so I'm just going to leave it there. so I'm just going to leave it there. It's like n multiplied by 1. It's like n multiplied by 1. Just like down here in the denominator, Just like down here in the denominator, that's going to be that's going to be 1 multiplied by n squared. 1 multiplied by n squared.
Okay, Okay, so now this will be 12 m to the so now this will be 12 m to the 6th, 6th, 6th I should say. 6th I should say. Now, Now, n over n squared, n over n squared, I can do this a couple of different ways. I can do this a couple of different ways. I can treat this as an exponent of 1 and then subtract the 2 there.
I can treat this as an exponent of 1 and then subtract the 2 there. So this is going to end up being So this is going to end up being 12 m to the 12m to the 6th n to the negative 1. 6th n to the negative 1. And I'm running out of room, And I'm running out of room, so I'll just go sideways here. so I'll just go, I'll go sideways here.
So this will be So this will be 12 m to the 6th all over n. 12m to the 6th all over n. Now that is not the only way to see that.
Now that is not the only way to see that. Okay, but if you're just getting comfortable with these ideas of negative exponents, But if you're just getting comfortable with these ideas of negative exponents, it might help to break it down like this. it might help to break it down like this.
Our next one. Okay, our next one. Okay, Okay, remember that that division symbol there puts brown.
remember that that division symbol there puts brackets around the numerator and the denominator. It's around the numerator and the denominator, so the very first thing I'm going to do is deal with what's in the numerator here. So the very first thing I'm going to do is deal with what's in the numerator here.
So x to the 5th multiplied by x to the negative 1 will be x to the 5th plus negative 1 over negative So x to the fifth multiplied by x to the negative one will be x to the fifth plus negative one. Over negative 2x to the negative 3. 2x to the negative 3. So we're going to get x to the 4 over negative So we're going to get x to the 4 over negative 2x to the negative 3. 2x to the negative 3. Oops, Oops, sorry. sorry. And now this is going to become, And now this is going to become, because I'm dividing powers of x there with the same base, because I'm dividing powers of x there with the same base, I'm going to subtract the exponents.
I'm going to subtract the exponents. Notice that that negative 2 isn't participating in any of this. Notice that that negative 2 isn't participating in any of this.
It's just going to stay there. It's just going to stay there. So this will become x to the 4 minus negative 3. So this will become x to the 4 minus negative 3 all over negative 2. All over negative 2, it just stays there.
It just stays there. And again, And again, I'm running out of room, I'm running out of room, so I'm going to have to go sideways here. so I'm going to have to go sideways here.
So this will be x to the 7 over negative 2. So this will be x to the 7 over negative 2. And I can put that negative in front. And I can put that negative in front. In fact, In fact, it looks a little bit like a it looks a little better to have the negative moved up into the numerator.
a little better to have the negative moved up into the numerator. When you've got a single factor of negative 1 in a fraction, When you've got a single factor of negative one in a fraction, yeah, yeah, you can move that kind of wherever you want. you can move that kind of wherever you want.
But it looks better in the numerator. But it looks better in the numerator. Oops. Oops.
Okay, Okay, next. next. Once again, Once again, I'm going to be pretty strict with the order of operations.
I'm going to be pretty strict with the order of operations. So the very first thing I'm going to do is focus on what's inside the brackets. The very first thing I'm going to do is focus on what's inside the brackets. Now, Now bearing in mind that that division statement there splits this up into parentheses there, bearing in mind that that division statement there splits this up into parentheses there, parentheses there. parentheses there.
But there's nothing that I can simplify in the numerator. But there's nothing that I can simplify in the numerator. There's nothing I can simplify in the denominator.
There's nothing I can simplify in the denominator. So that's as good as it's going to get. So that's as good as it's going to get.
So the next thing I'm going to do is deal with this division here. So the next thing I'm going to do is deal with this division here. And so now... And so now... The 5 and the 4, The 5 and the 4, I really can't do anything with to simplify that, I really can't do anything with to simplify that, so I'm just going to leave that.
so I'm just going to leave that. But this is going to become m to the 1 minus 2, But this is going to become m to the 1 minus 2, because division becomes subtraction. because division becomes subtraction.
Just like, Just like, whoops, oops, I haven't given myself... I haven't given myself enough room there. Enough room there.
Just like this will become n cubed minus 1 to the third minus 1 all squared. Just like this will become n cubed minus 1 to the third minus 1 all squared. And so this will become And so this will become 5m to the negative 5m to the negative 1. 1m, whoops, Whoops.
Never mind. I did that, Did that. weird, Weird.
n squared all over 4 squared. n squared all over 4 squared. And then that m to the negative 1 is going to drop into the denominator, And then that m to the negative 1 is going to drop into the denominator. so So 5n squared over 5n squared over 4m squared. 4m squared.
Now, Now, everything here is going to get squared. everything here is going to get squared. The numerator is going to get squared, The numerator is going to get squared, both terms. both terms. The denominator is going to get squared, The denominator is going to get squared, both terms.
both terms. So this is going to become 5 squared n squared squared. So this is going to become 5 squared n squared squared, so I'm going to multiply the two together. So I'm going to multiply the two together. That's a power of a power.
That's the power of a power over 4 squared m squared. Over 4 squared m squared. So we'll get 25 into the fourth over 16. So we'll get 25n to the fourth. Over 16 m squared.
m squared. And again, And again, you very likely could have gotten there faster than what I'm doing here. you very likely could have gotten there faster than what I'm doing here.
The more comfortable you get with this, The more comfortable you get with this, the quicker this process is going to get because you're going to see where you can... the quicker this process is going to get because you're going to see where you can it's not necessarily skipping steps. It's not necessarily skipping steps. I don't want you to think that I'm skipping steps because I'm not.
I don't want you to think that I'm skipping steps because I'm not. I wouldn't have been skipping steps. I wouldn't have been skipping steps. I can get here really quickly. I can get here really quickly.
Not because I'm skipping steps, Not because I'm skipping steps, but because the more comfortable you get with this, but because the more comfortable you get with this, the more steps you can do at once. the more steps you can do at once. You don't have to break them down into individual steps written out here. You don't have to break them down into individual steps written out here.
You can bunch them up here. You can bunch them up here. That's really what we want to get you to here.
That's really what we want to get you to here. Okay, Okay, following order of operations, following order of operations, remember that that line puts a set of parentheses there, remember that that line puts a set of parentheses there, so this is going to become m cubed n to the negative 2 all over. so this is going to become m cubed n to the negative 2 all over. Okay, Okay, well first thing I'm going to do here is I would try to simplify what's inside these brackets here, well first thing I'm going to do here is I would try to simplify what's inside these brackets here, but there's really nothing to simplify. but there's really nothing to simplify.
to simplify. So the very first thing I've got to do then down here is deal with the exponent. So the very first thing I've got to do then down here is deal with the exponent. So this is going to become So this is going to become 3 squared m squared. 3 squared m squared.
Now here's a spot where I'm just going to, Now here's a spot where I'm just going to, instead of writing this out, instead of writing this out, I'm just going to do this. I'm just going to do this. Because this is going to be n to the 4th squared, Because this is going to be n to the fourth squared. this will be n to the 8th. This will be n to the eighth.
And now I've still got m to the 5th and squared over here. And now I've still got m to the fifth and squared over here. So m, So m, here we go, here we go, m cubed, m cubed, n to the negative 2. n to the negative 2. This is going to become 9. This is going to become 9. Now, Now, and then again, and again, I'm going to try to speed this up a little.
I'm going to try to speed this up a little bit here. little bit here instead of writing this out I'm just going to do it this is going to be M squared multiplied by M to the fifth I just need to multiply those together here and that's C nice I wanted to write that out so I'm just gonna do it here Instead of writing this out, I'm just going to do it. This is going to be m squared multiplied by m to the 5th.
I just need to multiply those together here. And see, I wanted to write that out. It's so odd. I'm just going to do it here. 2 plus 5 is going to be 7 just like n to the 8th n squared is going to be n to the 10 now 2 plus 5 is going to be 7. Just like n to the 8th, n squared is going to be n to the 10. Now, That 9 is just going to stay down there in the denominator.
That 9 is just going to stay down there in the denominator. But for the time being, But for the time being, and I will do this kind of one step at a time here. and I will do this kind of one step at a time here.
Well, sort of. Sort of. I'm going to speed this up a little bit more than what I've been doing here.
I'm going to speed this up a little bit more than what I've been doing here. I've got m cubed now over m to the 7th. I've got m cubed now over m to the 7th.
In the numerator, In the numerator, that's going to be m to the 3 minus 7. that's going to be m to the 3 minus 7. So that's going to be m to the negative 4. So that's going to be m to the 7th. negative 4. Here, Here, this will be n to the negative 2 divided by n to the 10th is going to be, this will be n to the negative 2 divided by n to the 10th. This is going to be, in the numerator, in the numerator, will be n to the negative 2 minus 10 or n to the negative 12. will be n to the negative 2 minus 10 or n to the negative 12. And the very last thing I'm going to do, The very last thing I'm going to do, because the question was asking me to write these things with positive exponents, because the question was asking me to write these things with positive exponents, I'm going to move these guys into the denominator. I'm going to move these guys into the denominator.
So now, So now, I've seen people do this where they do this incorrectly. I've seen people do this where they do this incorrectly. They think they're moving into the denominator. They think they're moving into the denominator, denominator, and yet they don't write a fraction. and yet they don't write a fraction.
And if you don't write a fraction, And if you don't write a fraction, the assumption is that what you've done is put it in the numerator. the assumption is that what you've done is put it in the numerator. So if everything moves out down to the denominator, So if everything moves out down to the denominator, that's going to leave me with a 1 in the numerator, it's going to leave me with a 1 in the numerator.
9m to the 9m to the 4th, n to the 12th. 4th, n to the 12th. Okay, Okay, and finally, and finally, here, here, the negative 3. the negative 3, Now, now what I would try to do here first is simplify what's inside, what I would try to do here first is simplify what's inside, but there's nothing to simplify.
but there's nothing to simplify. The 2 and the 3, The 2 and the 3, that's 2 thirds, that's 2 thirds. That's about as simple as that fraction's going to get.
that's about as simple as that fraction's going to get, There's no shared. there's no shared factors there. factors there. x squared and y, x squared and y, as far as we can tell, As far as we can tell, there's nothing common about those two. there's nothing common about those two.
Okay, Okay, got to leave those separate here. got to leave those separate here. So what I'm going to do is I'm going to treat this as to the power of negative 1 times 3. So what I'm going to do is I'm going to treat this as to the power of negative 1 times 3. So I'm going to deal with the negative 1 first, So I'm going to deal with the negative 1 first, which is going to cause a reciprocal cubed.
which is going to cause a reciprocal cubed. Now, Now... I see a negative that's in here, I see a negative that's in here, and I'm cubing a negative, and I'm cubing a negative, so I already know that that negative is going to stay.
so I already know that that negative is going to stay. Now I'm just going to start to do these things really quickly here. Now, I'm just going to start to do these things really quickly here.
So this is going to become 3 cubed, So this is going to become 3 cubed, because they're multiplied together, because they're multiplied together, they're not added or subtracted here. they're not added or subtracted. subtracted here, so 3 cubed is 27. So 3 cubed is 27. y cubed is just that, y cubed is just that, y cubed.
y cubed. I already know the negative is going to pop out because a negative cubed is going to stay negative, I already know the negative is going to pop out, because a negative cubed is going to stay negative. But 2 cubed is going to be 8. but 2 cubed is going to be 8. And now here's the one that we need to make sure that we get right here.
And now here's the one that... we need to make sure that we get right here, x squared cubed, x squared cubed, if I've got a power of a power, if I've got a power of a power, you simply multiply the exponents and you get x to the sixth. you simply multiply the exponents and you get x to the sixth. And so there you go.
And so there you go. So, So, like I was saying, like I was saying, the more comfortable you get with this, the more comfortable you get with this, the more steps you can do simultaneously is going to be the best way to write that. the more steps you can do simultaneously is going to be the best way to write that. Actually, Actually, I say that. say that.
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