Understanding Algebraic Expressions Simplification

In this video, we're going to do a basic review of simplifying algebraic expressions. So let's say if you have the expression 5x plus 8 minus 2x plus 5. How can we simplify this expression? To do so, you need to combine like terms. We can't add 5x and 8. We can't say that's 13x because these are not like terms. 5x is similar to 2x because they carry the same variable x. So we can combine 5x and negative 2x. 5 plus negative 2 is the same as 5 minus 2, and that's 3x. Now, 8 and 5 are constants. They're like terms. So we can add those two numbers since they don't have a variable. 8 plus 5 is 13. And so the answer for this problem is 3x plus 13. So this expression is equivalent or it has the same value as this expression here. Let's try another problem. Simplify the following expression. 3x plus 5y. minus 9x plus 7y. So first, let's identify the terms that are similar to each other. So 3x and negative 9x are like terms. 5y and 7y are like terms, they're similar. So let's combine 3x and negative 9x. 3 minus 9 is negative 6. 5 plus 7 is 12. So this is the answer. Negative 6x plus 12y. Or we can write it like this as well. 12y minus 6x. These expressions are equivalent expressions. Here's another problem that you can work on. Simplify the following expression. 5xy plus 6x squared plus 8xy. minus 9x squared. So feel free to pause the video and work on this particular example. So let's begin by identifying which terms are like terms. 5xy and 8xy are like terms. They both have the variables xy. So 5 plus 8 is 13. So it's going to be 13xy. And 6x squared, negative 9x squared, they're like terms, 6 minus 9 is negative 3, so it's going to be negative 3x squared. So this is the answer. Now before we move on, another concept that you need to be familiar with is the distributive property. So let's say if we have 9 multiplied by 5x plus 4. To simplify this expression, we can distribute 9 to everything inside the parentheses. So 9 times 5x is 45x. And 9 times 4 is 36. So this expression is equal to 45x plus 36. Let's try another example using the distributive property. So try this one. 7 times 3x squared minus 8x plus 2. So let's distribute. 7 times 3 is 21. The x squared will carry over. 7 times negative 8x is negative 56x. And 7 times positive 2 is negative 56x. 2 is positive 14 so that's how you can use the distributive property to simplify expressions try this one simplify this expression 5 times 3x plus 4 minus 7x plus 8 so go ahead and simplify this First, let's distribute. 5 times 3x is 15x, and 5 times 4 is 20. And let's rewrite what we have on the right, negative 7x plus 8. So now, let's combine. like terms. So let's combine 15x and negative 7x. 15 minus 7 is 8 and we can combine the constants 20 and 8. 20 plus 8 is 28. So this is the answer. Let's work on another example. Let's try 7 times 2x minus 3 plus 4 times 9x minus 5. Go ahead and work on this example. Simplify this expression. Distribute and then combine like terms. So 7 times 2x is 14x, and 7 times negative 3 is negative 21. Positive 4 times 9x is positive 36x. Positive 4 times negative 5 is negative 20. Whenever you multiply two positive numbers, you will get a positive result. If you multiply a positive number and a negative number, you're going to get a negative number. And whenever you multiply two negative numbers, you're going to get a positive number. So make sure to remember those rules. Now let's combine like terms. So 14x plus 36x is equal to 50x. Negative 21 plus negative 20, which is the same as negative 21 minus 20. That's a negative 41 So this is the answer to this particular problem, so that's all you need to do Let's try another example 3x squared plus 5x minus 6 minus 4x squared plus 7x minus 3. So, if you're taking a test and you see a problem that looks like that, what would you do? Now, the first thing that we can do is we can get rid of the parentheses on the left side. We really don't need it. So basically, if you don't see a number in front, it's an invisible 1. And the same is true here. It's a negative 1 in front of it. So 1 times everything inside of the parentheses will be... whatever is inside of the parentheses. 1 times 3x squared is 3x squared. 1 times 5x is 5x, and 1 times negative 6 is negative 6, so we really don't need the parentheses on the left, because there's nothing in front of it except an invisible 1. Now, the negative sign is important for the parentheses on the right. If we multiply everything inside the parentheses by negative 1, it will have an impact. Negative 1 times 4x squared is negative 4x squared, so the sign is going to change. Negative 1 times 7x is negative 7x. And negative 1 times negative 3, two negative numbers that are multiplied to each other, will give you a positive number. So this is going to be positive 3. Now, let's combine like terms. 3x squared minus 4x squared is negative 1x squared, or simply negative x squared. 5x minus 7x is negative 2x. And negative 6 plus 3 is negative 3. So the final answer is negative x squared minus 2x minus 3. So that's how you can simplify a polynomial expression that looks like that. So this is called subtracting polynomials. So let's try another example. 5 times 2x cubed. plus 5x squared minus 8 minus 3 times 4x squared plus 5x plus 9. Go ahead, distribute, and then combine like terms. So let's multiply 5 by 2x cubed. So that's going to be 10x cubed. And then 5 times 5x squared. It's 25x squared. 5 times 5 is 25. And then 5 times negative 8 is negative 40. Now let's distribute negative 3 to everything on the right side. So negative 3 times 4 is negative 12. And negative 3 times 5 is negative 15 or negative 15x. And negative 3 times positive 9 is negative 27. So now what we need to do is combine like terms. So we only have 1x cubed, we have 2x squared, one term with x, and two constants. So we can't combine 10x cubed with anything, so let's simply rewrite it in the next line. So 10x cubed, now we can combine 25x squared and negative 12x squared. They're like terms 25 minus 12 is 13 now There's nothing to combine the negative 15x with so it's by itself Let's rewrite it and finally we can combine negative 40 and negative 27 which is negative 67 so this is the answer Now, what's going to happen if we multiply a monomial by another monomial? If you multiply x squared times x cubed, this is going to be x to the fifth power. Whenever you multiply a common variable, you need to add the exponents. So with that in mind, try these examples. x to the fourth times x to the seventh, x to the third times x to the negative eight. and x5 times x to the minus 3. So 4 plus 7 is 11. 3 plus negative 8 is negative 5. By the way, if you have a negative exponent, you can rewrite it like this. This is 1 divided by x to the positive 5. If you move the variable from the numerator to the denominator, that's the bottom of the fraction, the exponent will change sign from negative 5 to positive 5. Now what about the next one? x to the 5th power times x to the negative 3. If we add 5 and negative 3, that's the same as 5 minus 3, that's 2. So the answer is x squared. What if you saw a problem like this? 4x cubed times 5x squared. What would you do? So if you were to see this on a test, how would you answer it? The first thing we need to do is multiply the constants. 4 times 5. 4 times 5 is equal to 20. And then multiply the variables. x cubed times x squared. And that's x to the 5th power. So let's try some more examples. Try these. 3x to the 4th times 7x to the 5th. Let's rewrite that. And then 8x to the 3rd times 5x to the 6th. and 4x7 times 3x to the 5th. Try those examples. So 3 times 7 is equal to 21 and x to the 4th times x to the 5th is x to the 9th because 4 plus 5 is 9. So now let's move on to the next example. 8 times 5 is 40. x cubed times x to the 6th. 3 plus 6 is 9. So That's x to the 9. 4 times 3 is 12, and 7 plus 5 is also 12, so this is 12x to the 12th power. What about multiplying binomials with multiple variables? Try this one. So the principle is the same. Let's multiply the constants first. 8 times 7 is 56. x squared times x cubed is x to the 5th power, since 2 plus 3 is 5. y cubed times y to the 4th power is y to the 7th, since 3 plus 4 is 7. So let's try some examples. Try this one. 5x cubed, y to the 4th, times 7x squared, y to the 8th. And then 3xy squared times 5x cubed y. And also 4x to the fourth, y to the negative 3, times 5x to the third, y to the negative fifth. So 5 times 7 is 35. x cubed times x squared is x to the 5th power. And for y to the 4th times y to the 8th, 4 plus 8 is 12, so it's y to the 12th power. Now let's go ahead and make some space. So let's get rid of that stuff. And let's move this over here. 3 times 5 is 15. Now what is x times x cubed? If you don't see an exponent, there is an invisible 1. So it's really 1 plus 3, so it's x to the 4th. And here, this is y squared times y to the 1st power. 2 plus 1 is 3, so that's going to be y cubed. Now let's try the last example. So let's multiply 4 times 5, which is 20. And then x to the fourth times x to the third, that's x to the seventh power. and then y to the negative 3 times y to the negative 5. Negative 3 plus negative 5 is negative 8. But since we have a negative exponent, we can move it to the bottom. So it's 20x to the 7 divided by y to the 8th power. So we talked about multiplying monomials. We said that x to the fourth times x to the fifth is x to the nine because four plus five is nine. Now what about dividing monomials? Let's say if we want to divide x to the eight by x to the third. If you're going to divide, you need to subtract the exponents. Eight minus three is five. And that's basically what you need to do. So let's work on some examples. Try these. Well actually, let's just do this one first, and then I'll give you some more. So, for this one, it's basically 7 minus 4, which is 3, so it's x cubed. This is the answer. Now, another way in which you can see it, x to the 7 is basically 7x variables multiplied to each other. x to the fourth is basically four x variables multiplied to each other. You can cancel four x's, and notice that you'll be left with three x variables on top, which is x to the third. Let's try this example. x squared divided by x to the fifth power. So we need to subtract the top exponent by the bottom exponent. So this is going to be a 2 minus 5, which is negative 3. And whenever you have a negative exponent, you want to get rid of the negative sign. So if the negative... exponent is on top, move the x variable to the bottom, or to the denominator of the fraction. So this is going to be 1 over x cubed. Now another way in which you can see that is you can expand the expression. X squared is basically x. times X and X to the fifth power is X times X times X five times basically now we can cancel two X variables so if you cancel everything and if there's nothing on top this going to be a 1 because X divided by X is 1 notice that we have three X variables on the bottom So it's 1 over x cubed. As you can see, the answer is the same. So that's another way in which you could consider it when you're multiplying or dividing by monomials. If you're multiplying monomials, let's say x squared times x cubed. The reason why it's x to the fifth power is because x squared is x times x. x cubed is x times x times x. So you have five x variables. I meant to say 5x variables. It kind of rolled off my tongue in the wrong way, but that's x to the 5th power, as you can see. So let's try some more examples. Try these. x to the 9th divided by x to the 4th. x cubed over x to the 11th. x to the negative 4th divided by x to the 5th. x to the 3 divided by x to the negative 8, and x to the negative 7 divided by x to the negative 5. So let's start with the first one. This is going to be 9 minus 4, which is 5. So that's it for that example. And now, let's work on this one. So this is going to be the top number minus the bottom number, so 3 minus 11, which is negative 8. And we can rewrite that as 1 over x to the 8. So now let's try this one. So, this is going to be negative 4 minus 5, the top number minus the bottom number. You can put it in parentheses if that helps. Negative 4 minus 5 is negative 9, and we can rewrite that as 1 over x to the 9. Another way in which you could solve this problem, if you want to see it in a different way, you can... move the X variable from the top to the bottom if you do that the negative 4 will become positive 4 so this is 1 divided by X to the fourth times X to the fifth and we know that X to the fourth times X to the fifth is X to the ninth because 4 plus 5 is 9 so that's another way or a quicker way to get the same answer So let's try this one. This is going to be the top number 3 minus the bottom number, which is negative 8. And since we have two negatives next to each other, this is the same as 3 plus 8, which is 11. Now, another way in which we could solve it, we can take the x variable, move it to the top, and so this is going to be positive x to the 8, or x to the positive 8, which will still give us x to the 11th. So as you can see, there's more than one way to get the answer. Now for this one, we can subtract the exponents, the top minus the bottom, negative 7 minus negative 5, which is the same as negative 7 plus 5, which is negative 2. Negative 7 plus 5. is the same as 5 minus 7. That's negative 2, which is 1 over x squared. So that seems like a lot of work. Or we can do it this way. We can take the x to the negative 7 on the top and move it to the bottom so that the exponent will be positive. And the other x, the x to the negative 5, we can move it to the top. So this is x to the 5th divided by x to the 7th. x to the fifth is basically x times x times x five times. And x to the seven means that we have seven x variables on the bottom. So we can cancel five on top and five on the bottom, which means that we have two left over on the bottom. So x times x is x squared, so it's one over x squared. That's another way in which you can get the answer. Now what if you have examples like this? What's 36x to the 7th divided by 4x cubed? How would you simplify this expression? So first, divide the constants. 36 divided by 4 is 9. And then focus on the variables. x to the 7th divided by x cubed, that's going to be 7 minus 3, which is 4. And so, that's the answer for that one. Travis 128 X to the fifth Y to the fourth divided by 7 X to the 8 Y to the let's say negative 3 So let's divide 28 by 7 which is 4 and then 5 minus 8 is negative 3 that's for the x variable and for the y variable it's 4 minus negative 3 which is 4 plus 3 and that's 7 so it's y to the 7th. Now since we have a negative exponent we need to move it to the bottom. So this is 4y to the 7th divided by x cubed. So that's the answer. Now what if you have a fraction with two terms in the numerator? Let's say something like this and you're dividing it by a monomial or a single term. How can we simplify this expression? If you ever see an expression like this, you could separate the single fraction into two smaller fractions. So this is equivalent to 40x to the 5 divided by the 4x. and 12x squared divided by 4x so divide each term on top separately by 4x so 40 divided by 4 is 10 and 5 minus 1 is 4 so this is going to be 10x to the fourth power 12 divided by 4 is 3 and 2 minus 1 is 1 So it's 10x to the fourth plus 3x Now let's try another example like that So try this one 36x cubed plus 18 x squared Plus 15x divided by 3x. So this is equivalent to 36x cubed divided by 3x. plus 18x squared divided by 3x plus 15x divided by 3x. 36 divided by 3 is 12, and 3 minus 1 is 2, so this is going to be 12x squared. 18 divided by 3 is 6, 2 minus 1 is 1. In the last term, the x variables will cancel, so it's simply 15 divided by 3, which is 5, and this is the answer. So that's how you can simplify this expression. That's what you can do whenever you're dividing a polynomial, which has many terms, by a monomial, which has one term. Now what if we wanted to multiply a binomial which contains two terms by another binomial? How would you do it? So this process is called FOILING. We need to FOIL or multiply the two binomials. So let's multiply the first two terms on the outside, 3x times 2x, which is 3 times 2 is 6, x times x is x squared because 1 plus 1 is 2. 3x times negative 3 is negative 9x, 5 times 2x is 10x, and 5 times negative 3 is negative 15. All we need to do now is add like terms. So, negative 9x plus 10x, negative 9 plus 10, or 10 minus 9, is 1. 1x is simply x. So, this is the answer. 6x squared plus x minus 15. Now what if you have an expression that looks like this? What would you do? 5x minus 4 squared is equivalent to 5x minus 4 times 5x minus 4. It simply means that you have two of them. 5x times 5x is 25x squared. 5x times negative 4 is negative 20x. Negative 4 times 5x is negative 20x. And finally, negative 4 times negative 4 is positive 16. Whenever you multiply two negative numbers, you will get a positive number. So now we can add the two terms in the middle. Negative 20 plus negative 20 is negative 40x. So this is the answer. 25x squared minus 40x plus 16. Now, let's say if we wanted to multiply a trinomial. A trinomial is an expression with three terms. And let's multiply it by a binomial, which has two terms. So, 3 by 2, that should be 6. 3 times 2 is 6. So, initially, when we multiply these two expressions, we should have six terms. For the last example, when we multiplied a binomial by a binomial, it was a 2 by 2, so initially we got 4 terms before we simplified it. 3x squared times 3x is 9x cubed. 3 times 3 is 9. x squared times x to the first, it's going to be x cubed since 2 plus 1 is 3. 3x squared times negative 7 is negative 21x squared. 2x times 3x is 6x squared. And then 2x times negative 7 is negative 14x. 4 times 3x is 12x. And 4 times negative 7 is negative 28. So let's combine like terms. Negative 21 plus 6, or 6 minus 21, that's equal to negative 15x squared. Negative 14 plus 12, or 12 minus 14, that's negative 2x. And then everything else, let's bring it down. So it's 9x cubed minus 15x squared minus 2x minus 28. So this is the answer. Now let's multiply a trinomial by a trinomial. So 2x squared minus 7x plus 4 times 3x squared. plus 5x plus 7. So we're multiplying three terms by three terms. Initially we should get nine terms before we simplify, or before we combine like terms. So let's FOIL. 2 times 3 is 6. x squared times x squared is x to the fourth. Now what's 2x squared times 5x? 2 times 5 is 10. x squared times x is x cubed. And then 2x squared times 7 is 14x. Negative 7x times 3x squared, that's negative 21x cubed. Negative 7x times 5x is negative 35x squared. Negative 7x times 7 is negative 49x. 4 times 3x squared is 12x squared. 4 times 5x is 20x. And 4 times 7 is 28. Now I do need to make one small correction. 2x squared times 7 is 14x squared. I put 14x, so I just want to make sure that was corrected. Now 6x to the 4th is a term by itself. There's no other like terms similar to it, so we can rewrite it. now we only have two terms with an x cube that's 10x cubed and negative 21x cube so 10 minus 21 is negative 11 14x squared minus 35x squared that's negative 21 plus 12x squared that's negative 9x squared Negative 49x and 20x is negative 29x. And then finally, plus 28. So this is the answer. As you can see, whenever you multiply a trinomial by a trinomial, you should get nine terms. 1, 2, 3, 4, 5, 6, 7, 8, 9. And then after that, you can simplify by combining like terms.

We can't add 5x and 8. We can't say that's 13x because these are not like terms. 5x is similar to 2x because they carry the same variable x. So we can combine 5x and negative 2x. 5 plus negative 2 is the same as 5 minus 2, and that's 3x.

Now, 8 and 5 are constants. They're like terms. So we can add those two numbers since they don't have a variable. 8 plus 5 is 13. And so the answer for this problem is 3x plus 13. So this expression is equivalent or it has the same value as this expression here. Let's try another problem.

Simplify the following expression. 3x plus 5y. minus 9x plus 7y.

So first, let's identify the terms that are similar to each other. So 3x and negative 9x are like terms. 5y and 7y are like terms, they're similar.

So let's combine 3x and negative 9x. 3 minus 9 is negative 6. 5 plus 7 is 12. So this is the answer. Negative 6x plus 12y.

Or we can write it like this as well. 12y minus 6x. These expressions are equivalent expressions.

Here's another problem that you can work on. Simplify the following expression. 5xy plus 6x squared plus 8xy. minus 9x squared. So feel free to pause the video and work on this particular example.

So let's begin by identifying which terms are like terms. 5xy and 8xy are like terms. They both have the variables xy.

So 5 plus 8 is 13. So it's going to be 13xy. And 6x squared, negative 9x squared, they're like terms, 6 minus 9 is negative 3, so it's going to be negative 3x squared. So this is the answer.

Now before we move on, another concept that you need to be familiar with is the distributive property. So let's say if we have 9 multiplied by 5x plus 4. To simplify this expression, we can distribute 9 to everything inside the parentheses. So 9 times 5x is 45x.

And 9 times 4 is 36. So this expression is equal to 45x plus 36. Let's try another example using the distributive property. So try this one. 7 times 3x squared minus 8x plus 2. So let's distribute.

7 times 3 is 21. The x squared will carry over. 7 times negative 8x is negative 56x. And 7 times positive 2 is negative 56x.

2 is positive 14 so that's how you can use the distributive property to simplify expressions try this one simplify this expression 5 times 3x plus 4 minus 7x plus 8 so go ahead and simplify this First, let's distribute. 5 times 3x is 15x, and 5 times 4 is 20. And let's rewrite what we have on the right, negative 7x plus 8. So now, let's combine. like terms. So let's combine 15x and negative 7x.

15 minus 7 is 8 and we can combine the constants 20 and 8. 20 plus 8 is 28. So this is the answer. Let's work on another example. Let's try 7 times 2x minus 3 plus 4 times 9x minus 5. Go ahead and work on this example. Simplify this expression.

Distribute and then combine like terms. So 7 times 2x is 14x, and 7 times negative 3 is negative 21. Positive 4 times 9x is positive 36x. Positive 4 times negative 5 is negative 20. Whenever you multiply two positive numbers, you will get a positive result. If you multiply a positive number and a negative number, you're going to get a negative number. And whenever you multiply two negative numbers, you're going to get a positive number.

So make sure to remember those rules. Now let's combine like terms. So 14x plus 36x is equal to 50x.

Negative 21 plus negative 20, which is the same as negative 21 minus 20. That's a negative 41 So this is the answer to this particular problem, so that's all you need to do Let's try another example 3x squared plus 5x minus 6 minus 4x squared plus 7x minus 3. So, if you're taking a test and you see a problem that looks like that, what would you do? Now, the first thing that we can do is we can get rid of the parentheses on the left side. We really don't need it.

So basically, if you don't see a number in front, it's an invisible 1. And the same is true here. It's a negative 1 in front of it. So 1 times everything inside of the parentheses will be...

whatever is inside of the parentheses. 1 times 3x squared is 3x squared. 1 times 5x is 5x, and 1 times negative 6 is negative 6, so we really don't need the parentheses on the left, because there's nothing in front of it except an invisible 1. Now, the negative sign is important for the parentheses on the right.

If we multiply everything inside the parentheses by negative 1, it will have an impact. Negative 1 times 4x squared is negative 4x squared, so the sign is going to change. Negative 1 times 7x is negative 7x.

And negative 1 times negative 3, two negative numbers that are multiplied to each other, will give you a positive number. So this is going to be positive 3. Now, let's combine like terms. 3x squared minus 4x squared is negative 1x squared, or simply negative x squared. 5x minus 7x is negative 2x. And negative 6 plus 3 is negative 3. So the final answer is negative x squared minus 2x minus 3. So that's how you can simplify a polynomial expression that looks like that.

So this is called subtracting polynomials. So let's try another example. 5 times 2x cubed. plus 5x squared minus 8 minus 3 times 4x squared plus 5x plus 9. Go ahead, distribute, and then combine like terms. So let's multiply 5 by 2x cubed.

So that's going to be 10x cubed. And then 5 times 5x squared. It's 25x squared.

5 times 5 is 25. And then 5 times negative 8 is negative 40. Now let's distribute negative 3 to everything on the right side. So negative 3 times 4 is negative 12. And negative 3 times 5 is negative 15 or negative 15x. And negative 3 times positive 9 is negative 27. So now what we need to do is combine like terms. So we only have 1x cubed, we have 2x squared, one term with x, and two constants.

So we can't combine 10x cubed with anything, so let's simply rewrite it in the next line. So 10x cubed, now we can combine 25x squared and negative 12x squared. They're like terms 25 minus 12 is 13 now There's nothing to combine the negative 15x with so it's by itself Let's rewrite it and finally we can combine negative 40 and negative 27 which is negative 67 so this is the answer Now, what's going to happen if we multiply a monomial by another monomial?

If you multiply x squared times x cubed, this is going to be x to the fifth power. Whenever you multiply a common variable, you need to add the exponents. So with that in mind, try these examples.

x to the fourth times x to the seventh, x to the third times x to the negative eight. and x5 times x to the minus 3. So 4 plus 7 is 11. 3 plus negative 8 is negative 5. By the way, if you have a negative exponent, you can rewrite it like this. This is 1 divided by x to the positive 5. If you move the variable from the numerator to the denominator, that's the bottom of the fraction, the exponent will change sign from negative 5 to positive 5. Now what about the next one? x to the 5th power times x to the negative 3. If we add 5 and negative 3, that's the same as 5 minus 3, that's 2. So the answer is x squared.

What if you saw a problem like this? 4x cubed times 5x squared. What would you do? So if you were to see this on a test, how would you answer it? The first thing we need to do is multiply the constants.

4 times 5. 4 times 5 is equal to 20. And then multiply the variables. x cubed times x squared. And that's x to the 5th power.

So let's try some more examples. Try these. 3x to the 4th times 7x to the 5th.

Let's rewrite that. And then 8x to the 3rd times 5x to the 6th. and 4x7 times 3x to the 5th. Try those examples.

So 3 times 7 is equal to 21 and x to the 4th times x to the 5th is x to the 9th because 4 plus 5 is 9. So now let's move on to the next example. 8 times 5 is 40. x cubed times x to the 6th. 3 plus 6 is 9. So That's x to the 9. 4 times 3 is 12, and 7 plus 5 is also 12, so this is 12x to the 12th power. What about multiplying binomials with multiple variables? Try this one.

So the principle is the same. Let's multiply the constants first. 8 times 7 is 56. x squared times x cubed is x to the 5th power, since 2 plus 3 is 5. y cubed times y to the 4th power is y to the 7th, since 3 plus 4 is 7. So let's try some examples.

Try this one. 5x cubed, y to the 4th, times 7x squared, y to the 8th. And then 3xy squared times 5x cubed y. And also 4x to the fourth, y to the negative 3, times 5x to the third, y to the negative fifth. So 5 times 7 is 35. x cubed times x squared is x to the 5th power.

And for y to the 4th times y to the 8th, 4 plus 8 is 12, so it's y to the 12th power. Now let's go ahead and make some space. So let's get rid of that stuff.

And let's move this over here. 3 times 5 is 15. Now what is x times x cubed? If you don't see an exponent, there is an invisible 1. So it's really 1 plus 3, so it's x to the 4th.

And here, this is y squared times y to the 1st power. 2 plus 1 is 3, so that's going to be y cubed. Now let's try the last example.

So let's multiply 4 times 5, which is 20. And then x to the fourth times x to the third, that's x to the seventh power. and then y to the negative 3 times y to the negative 5. Negative 3 plus negative 5 is negative 8. But since we have a negative exponent, we can move it to the bottom. So it's 20x to the 7 divided by y to the 8th power. So we talked about multiplying monomials. We said that x to the fourth times x to the fifth is x to the nine because four plus five is nine.

Now what about dividing monomials? Let's say if we want to divide x to the eight by x to the third. If you're going to divide, you need to subtract the exponents. Eight minus three is five.

And that's basically what you need to do. So let's work on some examples. Try these. Well actually, let's just do this one first, and then I'll give you some more.

So, for this one, it's basically 7 minus 4, which is 3, so it's x cubed. This is the answer. Now, another way in which you can see it, x to the 7 is basically 7x variables multiplied to each other. x to the fourth is basically four x variables multiplied to each other.

You can cancel four x's, and notice that you'll be left with three x variables on top, which is x to the third. Let's try this example. x squared divided by x to the fifth power.

So we need to subtract the top exponent by the bottom exponent. So this is going to be a 2 minus 5, which is negative 3. And whenever you have a negative exponent, you want to get rid of the negative sign. So if the negative... exponent is on top, move the x variable to the bottom, or to the denominator of the fraction.

So this is going to be 1 over x cubed. Now another way in which you can see that is you can expand the expression. X squared is basically x. times X and X to the fifth power is X times X times X five times basically now we can cancel two X variables so if you cancel everything and if there's nothing on top this going to be a 1 because X divided by X is 1 notice that we have three X variables on the bottom So it's 1 over x cubed. As you can see, the answer is the same.

So that's another way in which you could consider it when you're multiplying or dividing by monomials. If you're multiplying monomials, let's say x squared times x cubed. The reason why it's x to the fifth power is because x squared is x times x. x cubed is x times x times x.

So you have five x variables. I meant to say 5x variables. It kind of rolled off my tongue in the wrong way, but that's x to the 5th power, as you can see. So let's try some more examples.

Try these. x to the 9th divided by x to the 4th. x cubed over x to the 11th. x to the negative 4th divided by x to the 5th.

x to the 3 divided by x to the negative 8, and x to the negative 7 divided by x to the negative 5. So let's start with the first one. This is going to be 9 minus 4, which is 5. So that's it for that example. And now, let's work on this one.

So this is going to be the top number minus the bottom number, so 3 minus 11, which is negative 8. And we can rewrite that as 1 over x to the 8. So now let's try this one. So, this is going to be negative 4 minus 5, the top number minus the bottom number. You can put it in parentheses if that helps. Negative 4 minus 5 is negative 9, and we can rewrite that as 1 over x to the 9. Another way in which you could solve this problem, if you want to see it in a different way, you can... move the X variable from the top to the bottom if you do that the negative 4 will become positive 4 so this is 1 divided by X to the fourth times X to the fifth and we know that X to the fourth times X to the fifth is X to the ninth because 4 plus 5 is 9 so that's another way or a quicker way to get the same answer So let's try this one.

This is going to be the top number 3 minus the bottom number, which is negative 8. And since we have two negatives next to each other, this is the same as 3 plus 8, which is 11. Now, another way in which we could solve it, we can take the x variable, move it to the top, and so this is going to be positive x to the 8, or x to the positive 8, which will still give us x to the 11th. So as you can see, there's more than one way to get the answer. Now for this one, we can subtract the exponents, the top minus the bottom, negative 7 minus negative 5, which is the same as negative 7 plus 5, which is negative 2. Negative 7 plus 5. is the same as 5 minus 7. That's negative 2, which is 1 over x squared. So that seems like a lot of work. Or we can do it this way.

We can take the x to the negative 7 on the top and move it to the bottom so that the exponent will be positive. And the other x, the x to the negative 5, we can move it to the top. So this is x to the 5th divided by x to the 7th. x to the fifth is basically x times x times x five times. And x to the seven means that we have seven x variables on the bottom.

So we can cancel five on top and five on the bottom, which means that we have two left over on the bottom. So x times x is x squared, so it's one over x squared. That's another way in which you can get the answer. Now what if you have examples like this? What's 36x to the 7th divided by 4x cubed?

How would you simplify this expression? So first, divide the constants. 36 divided by 4 is 9. And then focus on the variables.

x to the 7th divided by x cubed, that's going to be 7 minus 3, which is 4. And so, that's the answer for that one. Travis 128 X to the fifth Y to the fourth divided by 7 X to the 8 Y to the let's say negative 3 So let's divide 28 by 7 which is 4 and then 5 minus 8 is negative 3 that's for the x variable and for the y variable it's 4 minus negative 3 which is 4 plus 3 and that's 7 so it's y to the 7th. Now since we have a negative exponent we need to move it to the bottom.

So this is 4y to the 7th divided by x cubed. So that's the answer. Now what if you have a fraction with two terms in the numerator?

Let's say something like this and you're dividing it by a monomial or a single term. How can we simplify this expression? If you ever see an expression like this, you could separate the single fraction into two smaller fractions. So this is equivalent to 40x to the 5 divided by the 4x.

and 12x squared divided by 4x so divide each term on top separately by 4x so 40 divided by 4 is 10 and 5 minus 1 is 4 so this is going to be 10x to the fourth power 12 divided by 4 is 3 and 2 minus 1 is 1 So it's 10x to the fourth plus 3x Now let's try another example like that So try this one 36x cubed plus 18 x squared Plus 15x divided by 3x. So this is equivalent to 36x cubed divided by 3x. plus 18x squared divided by 3x plus 15x divided by 3x. 36 divided by 3 is 12, and 3 minus 1 is 2, so this is going to be 12x squared. 18 divided by 3 is 6, 2 minus 1 is 1. In the last term, the x variables will cancel, so it's simply 15 divided by 3, which is 5, and this is the answer.

So that's how you can simplify this expression. That's what you can do whenever you're dividing a polynomial, which has many terms, by a monomial, which has one term. Now what if we wanted to multiply a binomial which contains two terms by another binomial?

How would you do it? So this process is called FOILING. We need to FOIL or multiply the two binomials. So let's multiply the first two terms on the outside, 3x times 2x, which is 3 times 2 is 6, x times x is x squared because 1 plus 1 is 2. 3x times negative 3 is negative 9x, 5 times 2x is 10x, and 5 times negative 3 is negative 15. All we need to do now is add like terms.

So, negative 9x plus 10x, negative 9 plus 10, or 10 minus 9, is 1. 1x is simply x. So, this is the answer. 6x squared plus x minus 15. Now what if you have an expression that looks like this? What would you do? 5x minus 4 squared is equivalent to 5x minus 4 times 5x minus 4. It simply means that you have two of them.

5x times 5x is 25x squared. 5x times negative 4 is negative 20x. Negative 4 times 5x is negative 20x. And finally, negative 4 times negative 4 is positive 16. Whenever you multiply two negative numbers, you will get a positive number.

So now we can add the two terms in the middle. Negative 20 plus negative 20 is negative 40x. So this is the answer.

25x squared minus 40x plus 16. Now, let's say if we wanted to multiply a trinomial. A trinomial is an expression with three terms. And let's multiply it by a binomial, which has two terms.

So, 3 by 2, that should be 6. 3 times 2 is 6. So, initially, when we multiply these two expressions, we should have six terms. For the last example, when we multiplied a binomial by a binomial, it was a 2 by 2, so initially we got 4 terms before we simplified it. 3x squared times 3x is 9x cubed. 3 times 3 is 9. x squared times x to the first, it's going to be x cubed since 2 plus 1 is 3. 3x squared times negative 7 is negative 21x squared. 2x times 3x is 6x squared.

And then 2x times negative 7 is negative 14x. 4 times 3x is 12x. And 4 times negative 7 is negative 28. So let's combine like terms.

Negative 21 plus 6, or 6 minus 21, that's equal to negative 15x squared. Negative 14 plus 12, or 12 minus 14, that's negative 2x. And then everything else, let's bring it down. So it's 9x cubed minus 15x squared minus 2x minus 28. So this is the answer.

Now let's multiply a trinomial by a trinomial. So 2x squared minus 7x plus 4 times 3x squared. plus 5x plus 7. So we're multiplying three terms by three terms.

Initially we should get nine terms before we simplify, or before we combine like terms. So let's FOIL. 2 times 3 is 6. x squared times x squared is x to the fourth. Now what's 2x squared times 5x?

2 times 5 is 10. x squared times x is x cubed. And then 2x squared times 7 is 14x. Negative 7x times 3x squared, that's negative 21x cubed. Negative 7x times 5x is negative 35x squared. Negative 7x times 7 is negative 49x.

4 times 3x squared is 12x squared. 4 times 5x is 20x. And 4 times 7 is 28. Now I do need to make one small correction.

2x squared times 7 is 14x squared. I put 14x, so I just want to make sure that was corrected. Now 6x to the 4th is a term by itself.

There's no other like terms similar to it, so we can rewrite it. now we only have two terms with an x cube that's 10x cubed and negative 21x cube so 10 minus 21 is negative 11 14x squared minus 35x squared that's negative 21 plus 12x squared that's negative 9x squared Negative 49x and 20x is negative 29x. And then finally, plus 28. So this is the answer. As you can see, whenever you multiply a trinomial by a trinomial, you should get nine terms.

1, 2, 3, 4, 5, 6, 7, 8, 9. And then after that, you can simplify by combining like terms.

Transcript for:Understanding Algebraic Expressions Simplification

Transcript for:
Understanding Algebraic Expressions Simplification