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Order of operations example: fractions and exponents

The order of operations is essential for accurately evaluating complex math expressions. By following the steps of parentheses, exponents, multiplication and division, and finally addition and subtraction, you can simplify expressions and find the correct answer. Mastering these steps ensures a strong foundation in mathematics. Created by Sal Khan.

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Video transcript

- [Instructor] Pause this video and see if you can evaluate this expression before we do it together. All right, now let's work on this together. And we see that we have a lot of different operations here. We have exponents. We have multiplication. We have addition. We have division. We have parentheses. And so to interpret this properly, we just have to remind ourselves of the order of operations. So you start with parentheses, then go to exponents, then multiplication and division, then addition and subtraction. So we see that we're going to, whatever is over here, we're eventually going to square it. That's the only place that we have the parentheses. But how are we going to evaluate what's inside of these parentheses? So let's, then, think about, all right, we have an exponent here that we can evaluate. We know that 2 squared is the same thing as 2 times 2, which is the same thing as 4. No more exponents to evaluate. So then we go to multiplication and division. So we know by how this fraction sign is written that we need to evaluate the numerator and then divide it by the entire denominator right over here. Now in this numerator, we have to remind ourselves that we do this multiplication before we do this addition. We don't just go left to right. So we know that it's 1 plus, and I could put parentheses here to really emphasize that we do the multiplication first. So before this gets too messy, let me just rewrite everything. I'm going to do this multiplication up here first, and actually in the denominator I'm going to do this multiplication first as well. So this is all going to simplify to 1 over 14 or 1/14 times, Now this numerator here is going to be 1 plus 4 times 3. 4 times 3 is 12. All of that is going to be over 7 plus 2 times 3, which is of course equal to 6. And then I am going to have our plus 1 here, and then I square everything. Well now we can evaluate this numerator and this denominator. Find another color to do it in. This numerator, 1 plus 12, is going to be equal to 13. And 7 plus 6, interestingly, is also equal to 13. So we 1/14 or 1 divided by 14 times this whole thing squared, and inside you have 13 divided by 13 plus 1. Well, we know we need to do division before we do addition. So we will want to evaluate this part before we do the addition. What is 13 divided by 13? Well that's just going to be equal to 1. So I can rewrite this as 1 over 14 times 1 plus 1, all of that squared. And now we'll want to do this parentheses. So let's do that. 1 plus 1 is going to be equal, of course, to 2. And then we're going to do the exponents. 2 squared is, of course, equal to 4. And then we're going to multiply 1 over 14 times 4. Now you could interpret this, and they're equivalent. You could say, hey, this is the same thing as multiplying 1/14 times 4. Or you could say this is the same thing as multiplying 1 times 4 divided by 14. 1 times 4 divided by 14. Either way you look at it, you're going to get 4 over 14, and we're done. If you want, you could rewrite this by dividing both the numerator and the denominator by 2, and you could get 2 over 7. But that's how we can evaluate this pretty complex expression just step-by-step looking at what we can simplify first.