Welcome to Math with Mr. J. In this video, I'm going to cover the order of operations, and this is going to be a complete guide. We will start with why we use the order of operations, and then we will move on to a bunch of example problems. All of the different types of examples are put into chapters and time-stamped. Check the description for the full list.
So let's start with why we use the order of operations. Remember, the order of operations are parentheses, exponents, multiplication and division, and then addition and subtraction, PEMDAS. Now, simply put, we use the order of operations. That way, everyone is on the same page as far as working through problems. We all work through them the same way in order to get to the same answers.
and there isn't any confusion about how to work through things. We all use the same set of rules, the same set of instructions. That's the order of operations. Let's take a look at an example to show why this is important.
We have 12 minus 5 times 2. Let's work through this two different ways. We will start by forgetting about the order of operations. Let's say that a person works from left to right, starting with 12 minus 5. So what's this going to look like?
Well, 12 minus 5 is 7. Bring down everything we did not use, so the multiplication sign and the 2. Now we have 7 times 2, which is 14. So our final answer when we work from left to right is 14. Now let's do 12 minus 5 times 2 another way. We're going to start by doing multiplication first. So let's say this person starts with multiplication, which would be 5 times 2. So let's start there. 5 times 2 is 10. Bring down everything we did not use. So the 12 and the subtraction sign.
So now we end with 12 minus 10, which is... 2. So our final answer when we start with multiplication is 2. So you can see that we have two different answers, even for a simple problem like this. So imagine a more complex problem and the number of possible answers. That's why we need the order of operations, a set of rules we can all follow to get to the same answer. Now taking a look at the order of operations.
we do multiplication before subtraction. So two is the correct answer for this problem. Think of it like this. If everyone had their own set of rules for stoplights, we would have some issues.
So if some people used red for go and green for stop, and then others used green for go and red for stop, that would definitely cause confusion. But that's not how things work. We have a set of rules to go by.
That way we are... all on the same page. Now we can also relate this to measurement. If everyone had their own idea of what a foot, a meter, or any other unit of measure was, that wouldn't work out. There would be a lot of confusion and differences.
A foot, a meter, and all other units of measure are the same for everyone. That way we can all be on the same page and there isn't confusion from person to person. We have a standard way of doing things, a set of rules we can all go by. And that's what the order of operations is. So that's why we use the order of operations.
Let's move on to some example problems and how to use the order of operations. Here are the first two examples. And remember, as far as the order of operations, we start with parentheses.
So parentheses are priority. number one. If we see parentheses in an expression, we start there. Then we have exponents, then multiplication and division.
Now I do want to mention multiplication and division are on the same level. They are the same priority in the order of operations. So if we have both, we work from left to right.
And then addition and subtraction. Now, addition and subtraction are on the same level. They are the same priority.
So if we have both, we work from left to right. This will all make a lot more sense as we go through our examples. Just think of the order of operations like a set of instructions that we follow step by step. Now we have an acronym that we can think of in order to remember that order. PEMDAS.
So parentheses, exponents, multiplication and division, and then addition and subtraction. So PEMDAS just represents the order of operations. Let's jump into our examples and see exactly how all of this works.
Starting with number one, where we have 30 divided by and then in parentheses 13 minus 8. So let's work through the order of operations. Do we have any parentheses in this expression? Yes. So we start there.
We have 13 minus 8 in parentheses. 13 minus 8 is 5. Then we need to bring down everything we did not use. So we have 30 and then divided by 5. So now we have 30 divided by 5. We only have one operation here, division.
So that's what we need to do. 30 divided by 5 is 6. So our final answer, 6. So for number 1, we worked through the order of operations. We started with parentheses, then we brought down everything we did not use, and we ended with 30 divided by 5, which gave us 6. Let's move on to number 2, where we have 16 minus 5 times 3 plus 12. Let's work through the order of operations.
Do we have any parentheses? No. So let's move on to exponents.
Do we have any exponents? No. So let's move on to multiplication and division. Do we have any multiplication or division?
Yes. So that's where we start. We have multiplication.
We have 5 times 3. That is 15. Now we need to bring down everything we did not use. So we have 16 minus 15 and then plus 12. So we have 16 minus 15 plus... 12. And we need to continue to work through the order of operations. Any parentheses?
No. Any exponents? No.
Any multiplication or division? No. Any addition or subtraction?
Yes. We have both addition and subtraction. Since we have both addition and subtraction, we need to work from left to right. Addition and subtraction are on the same level. They are the same priority.
in our order of operations. So again, we need to work left to right. When working from left to right, we need to do subtraction first here.
So 16 minus 15 is 1. Bring down everything we did not use. So plus 12. And now we have 1 plus 12. We only have one operation left. So that's what we need to do.
We need to add. 1 plus 12 is 13. And that is our final answer. Let's move on to numbers three and four. Here are numbers three and four. Let's start with number three where we have seven squared minus 14 times two.
Let's work through the order of operations. Are there any parentheses in this expression? No.
So let's move on to exponents. Are there any exponents in this expression? Yes. So let's start there.
We have seven squared. which means 7 times 7, that's 49. Bring down everything we did not use. So now we have 49 minus 14 times 2. Let's continue to work through the order of operations.
Any parentheses? No. Any exponents? No. Any multiplication or division?
Yes, we have multiplication. So that's what we need to do next. We have 14 times 2, which is 28. Bring down everything we did not use.
So 49 minus 28. We have one operation left. Subtraction. So we need to subtract 49 minus 28 is 21. So our final answer, 21. Let's move on to number four, where we have 18 divided by, and then in parentheses, 6 plus 3, and parentheses times 15. Let's work through the order of operations. Any parentheses in this expression? Yes.
So we need to start there. We have 6 plus 3 in parentheses. That's 9. And then we need to bring down everything we did not use.
So now we have 18 divided by 9 times 15. Any parentheses? No. Any exponents?
No. Any multiplication or division? Yes, we actually have both multiplication and division.
Since we have both, we need to work from left to right. Multiplication and division are on the same level, so to speak. They are the same priority.
So when that happens, we work from left to right. When working from left to right, division comes first. We have 18 divided by 9. So let's do that.
18 divided by 9 is 2. Bring down everything we did not use, so times 15 there, and now we have 2 times 15. 2 times 15 gives us 30. Our final answer, 30. So there are four introductory examples. Let's move on to some examples that involve parentheses and exponents. Here are our examples involving parentheses and exponents.
Let's jump into number one where we have 13 plus 5 and then in parentheses 2 cubed plus 4 and parentheses divided by 2. Now one thing I want to mention before we get started, whenever we have a number next to parentheses, so we have 5 and then a set of parentheses, that's multiplication. So we have 5 times whatever we get within those parentheses. So let's go through the order of operations.
Are there any parentheses? Yes. So we start there. We need to do anything inside of parentheses.
Well, we have 2 cubed plus 4. So we do exponents before addition. So let's do 2 cubed first. 2 cubed means 2 times 2 times 2. 2 times 2 is 4. Times 2 is 8. So 2 cubed is 8. So in parentheses, we now have 8 plus 4. Now we can bring down everything we did not use.
So 13 plus 5, and then in parentheses, 8 plus 4, end parentheses, and then we have divided by 2. So let's go through the order of operations. Any parentheses? Yes. So we need to do anything in parentheses here.
We have 8 plus 4, which is... 12. Now we can bring down everything we did not use. So we have 13 plus 5 times 12. Whenever you have a number next to parentheses, that's multiplication.
So that's 5 times 12 divided by 2. So let's continue to work through this problem step by step. So any parentheses, anything within parentheses that we need to do? No.
Any exponents? No. Any multiplication or division? Yes, we have both. We have multiplication and division.
Multiplication and division are the same priority. So what we do, we work from left to right if we have both. So let's do 5 times 12 first. 5 times 12 is 60. Bring down everything we did not use. So 13. plus 60 divided by 2. So now we have 13 plus 60 divided by 2. Do we have any parentheses?
No. Any exponents? No.
Any multiplication or division? Yes, we have division. 60 divided by 2. So let's do that next. 60 divided by 2 is 30. Bring down everything we did not use. So we have 13 plus 30. All we have left is addition, so that's what we end with.
13 plus 30 gives us 43. And that is our final answer. So again, we just broke that problem down. We worked through that problem step by step following the order of operations.
Let's move on to number two where we have 15 minus 7 in parentheses, and that is squared minus. in parentheses 36 divided by 9, end parentheses, times 10. So let's work through the order of operations. Do we have any parentheses? Yes, we have two sets of parentheses.
So let's work from left to right. We will start with 15 minus 7. 15 minus 7 is 8. Now we can bring down everything we did not use. So we are squaring whatever we got within those parentheses. So bring down the exponent of 2. Then we have minus.
and then in parentheses 36 divided by 9, end parentheses, times 10. So now we have 8 squared minus, and then in parentheses 36 divided by 9, end parentheses, times 10. So let's continue to work through the order of operations. Do we have anything within parentheses? Yes, we have 36 divided by 9. So let's do that next.
36 divided by 9 is 4. Bring down everything we did not use. So 8 squared minus 4 times 10. So now 8 squared minus 4 times 10. Any parentheses? No.
Any exponents? Yes. So that's what we need to do next. We have 8 squared, which means 8 times 8. 8 times 8 is 64. Now we can bring down everything we did not use. So minus 4 times 10. Now we have 64 minus 4 times 10. Any parentheses?
No. Any exponents? No.
Any multiplication or division? Yes. We have 4 times 10, so we need to do that next.
4 times 10 is 40. Bring down everything we did not use. So we have 64 minus... 40 and we're only left with subtraction here.
So this is what we end with 64 minus 40 is 24 and that is our final answer So there are some examples involving parentheses and exponents Let's move on to examples that do not have parentheses or exponents Now we are going to take a look at some problems without parentheses or exponents. Let's jump into number one where we have 20 minus 4 times 4 plus 15 minus 6. So let's work through the order of operations. Do we have any parentheses?
No. Any exponents? No. Any multiplication or division? Yes.
So that's where we need to start. We have multiplication right here, 4 times 4. That gives us 16. Now we can bring down everything we did not use. And now we have 20 minus 16 plus 15 minus 6. So now we can continue to work through the order of operations.
Any parentheses? No. Any exponents?
No. Any multiplication or division? No. We... only have addition and subtraction left.
Addition and subtraction are on the same level, so to speak. They are the same priority. So since we have both addition and subtraction, we work from left to right. When working from left to right, we have subtraction first.
We have 20 minus 16. So let's do that next. 20 minus 16 is 4. bring down everything we did not use, and we can continue to work through the order of operations. So we have 4 plus 15 minus 6. So we can continue to work from left to right since we only have addition and subtraction left. When working from the left we have addition next, 4 plus 15, that gives us 19. Bring down everything we did not use, so minus 6 there, and now we have 19 minus 6, which we end with. 19 minus 6 gives us a final answer of 13. Let's move on to number 2, where we have 7 times 2 plus 9 divided by 3 minus 10 minus 7. So let's work through this using the order of operations.
Any parentheses? No. Any exponents?
No. Any multiplication or division? Yes, we have both multiplication and division. Multiplication and division are the same priority. So since we have both, we work from left to right.
When working from left to right, multiplication comes first. We have 7 times 2, which gives us 14. Bring down everything we did not use, and we can go from there. So now we have 14 plus 9. divided by 3 minus 10 minus 7. We have addition, division, subtraction, and subtraction. So division is going to come next.
So let's do 9 divided by 3, which is 3. Bring down everything we did not use. Now we have 14 plus 3 minus 10. minus 7. And again addition and subtraction are the same priority. So let's work from left to right. When working from left to right addition comes first. We have 14 plus 3. That gives us 17. Bring down everything we did not use and now we have 17 minus 10 minus 7. So we just have subtraction left.
So let's continue to work from left to right. So next, let's do 17 minus 10, which gives us 7. Bring down everything we did not use. And now we have 7 minus 7. And that's what we end with. 7 minus 7 gives us a final answer of 0. So there are some examples without parentheses or exponents. let's move on to examples that have parentheses, brackets, and braces.
Now we are going to take a look at some problems with multiple grouping symbols, parentheses, brackets, and braces. So taking a look at parentheses, brackets, and braces, we have parentheses, brackets, and then braces. Now, all of these are grouping symbols and take priority within the order of operations.
They come first. You can think of brackets and braces just like parentheses. Again, they are grouping symbols just like parentheses. Let's take a look at our examples and see how to work through problems involving multiple grouping symbols.
And you may hear this called nested parentheses. Nested parentheses just means that parentheses are inside of other grouping symbols. So grouping symbols inside of grouping symbols.
Let's jump into number one where we start with brackets and then within the brackets we have 28 minus and then in parentheses 6 plus 4 end parentheses and then we end the brackets divided by 2. Let's work through the order of operations. Do we have any parentheses or other grouping symbols? Yes, we have parentheses and brackets.
So we have parentheses nested within the brackets. When we have a problem like this with multiple grouping symbols, we always start with the innermost group. These parentheses right here, so 6 plus 4, this is the innermost group.
So this is where we need to start. 6 plus 4 is 10. Now we need to bring down everything we did not use. So we have 28 minus 10 in brackets divided by 2. So now we have brackets 28 minus 10 end brackets divided by 2. And now we can continue to work through the order of operations. So do we have any parentheses or other grouping symbols? Yes, we have brackets.
So that's what we need to do next. We have 28 minus 10 within those brackets. 28 minus 10 is 18. So we have 18 and then bring down everything we did not use.
So divided by 2 and we end with 18 divided by 2. That is 9. And... this is our final answer. Final answer, 9. Let's move on to number 2 where we have 50. minus and then braces, then brackets, five times, and then parentheses, seven plus one, end parentheses, end brackets, plus three squared, end braces.
We have multiple grouping symbols here, parentheses, brackets, and braces. So we need to start with the innermost group. That's going to be these parentheses right here. We have 7 plus 1 within those parentheses.
So that's where we start. 7 plus 1 is 8. Bring down everything we did not use. So we have 5 times 8 within brackets plus 3 squared.
And this is all within braces. And then we have 50 minus whatever we get. within those braces. Now we have 50 minus and then braces then brackets 5 times 8 and brackets plus 3 squared and braces. Let's continue to work through the order of operations.
Do we have any parentheses or other grouping symbols? Yes we have both brackets and braces so we need to do the innermost group next. That's going to be the brackets. we have 5 times 8 within the brackets. 5 times 8 is 40. Bring down everything we did not use, so plus 3 squared, and this is within braces.
Then we have 50 minus whatever we get within those braces. Now we have 50 minus, and then braces, 40 plus 3 squared and braces. So let's continue to work through the order of operations.
Do we have any parentheses or other grouping symbols? Yes, we still have those braces. So that's what we need to work within next. We have 40 plus 3 squared within the braces.
So we have addition and an exponent. Exponents come before addition. So we need to do 3 squared next. 3 squared means 3 times 3. So 3 squared... is 9. So we have 9 and then 40 plus 9 and this is within braces still and then we have 50 minus whatever we get within those braces.
Now we have 50 minus and then braces 40 plus 9 and braces. So we need to do 40 plus 9 next. 40 plus 9 is 49. Bring down everything we did not use and we now have 50 minus 49 and that's what we end with.
50 minus 49 is 1 and this is our final answer. Final answer 1. So there are some examples involving multiple grouping symbols. Let's move on to examples involving a fraction bar.
Now we are going to take a look at problems involving fraction bars. Now when it comes to fraction bars, they tell us to group everything together on top, the numerator, and then group everything together on the bottom, the denominator. The fraction bar acts as a grouping symbol. Group the top, group the bottom, and then lastly we divide the top by the bottom. So the numerator divided by the denominator.
Let's jump into our examples and see exactly what this all means. Starting with number 1 where we have 10 plus 5 times 6 above the fraction bar, so the numerator, and then 12 divided by 3 below the fraction bar, so the denominator. So that fraction bar tells us that we take the result of the top, the numerator, and divide it by the result of the bottom. the denominator. So we can think of it like this.
We are grouping the top and grouping the bottom. So group the numerator and group the denominator. Once we have those results, we divide.
Now as we work through this, we need to use the order of operations. Let's start with the top. We have 10 plus 5 times 6. So we need to multiply first and then add.
Remember, Multiplication comes before addition. Five times six gives us 30. So we have 10 plus 30. That gives us 40. So the numerator is 40. As far as the bottom, we have 12 divided by 3, which is 4. So the denominator is 4. And we end up with 40 over 4. Now we need to divide. We have our numerator, 40, divided by our denominator, 4. 40 divided by 4 gives us 10. So this is our final answer, 10. Now before we move on to number two, I do wanna mention we can work through problems like this that involve a fraction bar to the side.
So let me rewrite the problem here and go through what this will look like. So 10 plus five times six, and then we have the fraction bar, and then 12 divided by three. So let's start with the top. We need to multiply first. 5 times 6 is 30. So 10 plus 30 gives us 40. And then as far as the bottom, 12 divided by 3 gives us 4. So we end up with 40 over 4. So 40 divided by 4 now gives us that answer of 10. So just a different way to work through the problem.
We get the same answer either way, but if you prefer working your way down, or maybe you prefer working your way to the side, either way will work. Let's move on to number two. So now we can take a look at number two, where we have 28 minus 22 plus 15 on top, and then below the fraction bar we have in parentheses 27 minus 9 end parentheses divided by and then in parentheses 3 times 2 end parentheses.
Let's start with the top. So we have subtraction and addition which are on the same level. They are the same priority.
That means we work from left to right. So working from left to right we have subtraction first here. So 28 minus 22. That gives us 6. Then we have plus 15. So 6 plus 15, that gives us 21. So 21 is the numerator.
Now let's take a look at the bottom. So the denominator here. And we have two sets of parentheses. So we have 27 minus 9 and 3 times 2. For the bottom, we're going to work through this in a couple of steps, since this has a little more to it. Let's work through the parentheses.
So we have 27 minus 9. That gives us 18. Divided by. And then in the other set of parentheses, we have 3 times 2. That gives us 6. So we have 21 for the numerator. And then as far as the denominator, the bottom, let's do 18 divided by 6. That gives us 3. So we end up with 21 over 3. 21 divided by 3. That gives us 7. And this is our final answer. Let's work through this problem to the side as well to show what that will look like. So we have 28 minus 22 plus 15. Then we have the fraction bar, 27 minus 9. in parentheses, divided by 3 times 2 in parentheses.
Let's start with the top. So we have 28 minus 22. That gives us 6 plus 15 gives us 21. Now for the bottom, and we will do each set of parentheses. We will start with 27 minus 9. That gives us 18. divided by and then the other set of parentheses 3 times 2 gives us 6. So we have 21 over and then 18 divided by 6 is 3 and we end with 21 divided by 3 which gives us a final answer of 7. And again, it doesn't matter which way you go about working through these problems.
You can work your way down or you can work your way to the side. Either way will work as long as you follow the order of operations. And just remember, that fraction bar is a grouping symbol. We group the top and then we group the bottom. And then lastly, we divide the results.
So the result of the top divided by the result of the bottom. The numerator divided by the denominator. So there are some examples involving a fraction bar.
Let's move on to another example that is a little more complex. Now we are going to take a look at a more complex problem involving a fraction bar. Let's jump into our example, and we will start above the fraction bar here.
So we have in parentheses 6 squared minus 9, end parentheses, divided by, and then in parentheses, 7 plus 2, end parentheses, times 11. And then as far as below the fraction bar, we have in parentheses 18 minus 16, end parentheses, and... That is to the power of 4. And then we have minus 5. So again, that fraction bar tells us to take the result of the top and divide it by the result of the bottom. So the numerator divided by the denominator.
Let's work through this problem starting with the top. So do we have any parentheses? Yes.
So we need to start there. And I'm going to work to the side. for this example. So we have 6 squared minus 9. Let's start there. 6 squared means 6 times 6. So this is 36. So we have 36 minus 9. That gives us 27 divided by and the other set of parentheses we have 7 plus 2. That gives us 9 and then we have times 11. So we worked through the parentheses as far as the top goes.
Now, as far as the bottom, do we have any parentheses? Yes, we have 18 minus 16 within parentheses. 18 minus 16 gives us 2. And that is going to be to the power of 4. So we are taking what we get within those parentheses, and that is going to be to the power of 4. So 2 to the power of 4. minus 5. So let's continue to work through this problem using the order of operations.
Let's take a look at the top. So we have 27 divided by 9 times 11. We have both multiplication and division. Those are the same priorities, so we need to work from left to right.
When working from left to right, division comes first. So we have 27 divided by 9. That gives us 3, and then we have times 11. So 3 times 11 is 33. Now as far as the bottom, we have 2 to the power of 4 minus 9. So let's do 2 to the power of 4. Exponents come next. So 2 to the power of 4 means 2 times 2 times 2 times 2. 2 times 2 is 4 times 2 is 8 times 2 is... 16. So we have 16 minus 5. For the top we still have 33 and then for the bottom we end with subtraction.
16 minus 5 gives us 11. So we end up with 33 over 11. So now we divide. We have 33 divided by 11. The numerator, the top number, divided by the denominator, the bottom number. 33 divided by 11 is 3. So our final answer, 3. So there's another example involving a fraction bar. Let's move on to an example that involves a fraction bar, and the answer works out to be a fraction.
Now we are going to take a look at a problem involving a fraction bar, and the answer works out to be a fraction. Let's jump into our example where we have above the fraction bar 5 squared minus and then in parentheses 10 plus 9 end parentheses and then below the fraction bar in parentheses we have 11 minus 8 end parentheses times in parentheses 4 plus 2 end parentheses. Now let's start with the top. And we will work to the side here.
So do we have any parentheses? Yes. So let's start there. We have 10 plus 9 in parentheses, which is 19. So we have 5 squared.
minus 19. And then as far as the denominator below the fraction bar, do we have any parentheses? Yes, we have two sets of parentheses. Let's start with the first set where we have 11 minus 8. That gives us 3. So we have 3 times and then the second set of parentheses we have 4 plus 2. That gives us 6. So now we have 3 times 6 below the fraction bar. Now we need to continue to work through the order of operations. So taking a look at the top, we have 5 squared minus 19. We need to do 5 squared next.
We need to take a look at that exponent. 5 squared means 5 times 5. So that is 25. So now we have 25 minus 19 above the fraction bar and then below the fraction bar we have 3 times 6 and that is 18. So we have 25 minus 19 over 18. Now we can subtract as far as the top goes. So 25 minus 19 is 6. So we end up with 6 over 18. 6 18ths. Now 6 18ths is our final answer. We have a proper fraction here.
The numerator is less than the denominator and that's okay. When this happens we can always look to see if we can simplify the fraction. In this case we can.
We have a greatest common factor other than 1 that we can divide both 6 and 18 by. 6 is actually the greatest. common factor. So let's divide 6 and 18 by 6 in order to simplify. So 6 divided by 6 gives us 1 and then 18 divided by 6 gives us 3 and we end up with 1 third.
The only common factor between 1 and 3 is 1 so this is in simplest form. Final simplified answer, one third. So there is an example where the answer works out to be a fraction.
Let's move on to examples involving positive and negative integers. Now we are going to take a look at problems involving positive and negative integers. Let's jump into our examples, starting with number one, where we have negative 18 divided by negative.
2 times and then in parentheses negative 11 plus 6 end parentheses. So let's work through the order of operations in order to break this problem down. Do we have anything to do with in parentheses?
Yes, we have negative 11 plus 6. So let's start there. Negative 11 plus 6 gives us negative 5. So we start. at negative 11 and add 6. So we are increasing in value by 6 and get negative 5. So now we can bring down everything we did not use and go from there. Now we have negative 18 divided by negative 2 times negative 5. And you'll notice I'm keeping some of the negatives within parentheses. That helps us stay a little more organized and we don't confuse any negatives for subtraction or forget any of the negative signs.
It makes the negatives a little more clear and again helps us stay organized. I left the negative 18 without parentheses because it's in the front and the negative is clear, but we can definitely put it. in parentheses as well if we would like. So let's continue to work through the order of operations.
Anything to do with in parentheses? No. Any exponents?
No. Any multiplication or division? Yes. We actually have both multiplication and division.
Multiplication and division are the same priority, so we work from left to right. When working from left to right, we have division first. We have... Negative 18 divided by negative 2. So we have a negative divided by a negative.
That gives us a positive. So negative 18 divided by negative 2 is 9, a positive 9. Then we can bring down everything we did not use. So times negative 5. And now we have 9 times negative 5. A positive times a negative equals a negative. So 9 times negative 5 is negative 45. And this is our final answer, negative 45. Let's move on to number two, where we have in parentheses negative four minus four end parentheses. And we are squaring what we get within those parentheses plus negative three cubed divided by nine.
So let's work through the order of operations. Do we have anything to work through within parentheses? Yes, we have negative 4 minus 4. So let's start there.
Now remember, one thing we can do when we have subtraction, we can add the opposite. And that can help when we have problems involving negatives. So let's add the opposite here.
We have negative 4 plus, and then the opposite of positive 4 is negative 4. So negative 4 plus negative 4 or negative 4 minus 4, that gives us negative 8. And we need to keep that negative 8 within parentheses because we are squaring this result. We are squaring what we got within the parentheses. It's very important to keep that negative result, negative 8, in parentheses. Because again, we are squaring it. We have an exponent.
Now remember, and I'm going to come to the side here, negative 8 squared is different than if we do not have the parentheses. Negative 8 squared with parentheses means negative 8 times negative 8, which equals 64. The negative is included with the exponent. Without the parentheses, that means 8 times 8, which is 64. The exponent is only applied to the 8, and then we have the negative sign in front.
So without the parentheses, that equals negative 64. Now if you need more of an explanation as far as the difference between those, I go into more detail in another video. That link is in the description. So we have negative 8 squared plus negative 3 cubed divided by 9. So now we have negative 8 squared plus negative 3 cubed divided by 9. Do we have anything to work through within parentheses?
No. Do we have any exponents? Yes, we have two exponents. So let's start with negative 8 squared, which means negative 8 times negative 8. So a negative times a negative.
A negative times a negative equals a positive. So this gives us positive 64. And we will go one step at a time here. So let's bring down everything we did not use. And now we have 64 plus... negative 3 cubed divided by 9. So now we can do negative 3 cubed.
That means negative 3 times negative 3 times negative 3. Negative 3 times negative 3 is positive 9. So we have positive 9 times negative 3. That gives us negative 27. So we have negative 27. Bring down everything. we did not use and now we have 64 plus negative 27 divided by 9. Now just as a quick recap as far as negative 3 cubed and how we got negative 27 I'm going to come to the side here so we had negative 3 cubed which means negative 3 times negative 3 times negative 3. Negative 3 times negative 3 is positive 9. And then we had positive 9 times negative 3. A positive times a negative equals a negative. So 9 times negative 3 equals negative 27. So now as far as the order of operations, we have 64 plus negative 27 divided by 9. Do we have anything within parentheses that we need to work through?
No. Do we have any exponents? No. Do we have any multiplication or division?
Yes. So that's what we do next. We have negative 27 divided by 9. So a negative divided by a positive. That gives us a negative. So negative 27 divided by 9 gives us negative 3. Bring down everything we did not use.
And we end with 64 plus negative 3, which is 61. And this is our final answer. So there you have it. There is a complete guide to the order of operations. I hope that helped. Thanks so much for watching.
Until next time, peace.