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Order of Operations: Exponents and Radicals

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Today, we're going to talk about exponents and radicals and how they fit into the order of operations. Radicals can also be written as exponents, and exponents can be written as radicals. So we actually look at them together when we're doing PEMDAS, or the order of operations. They both fall under exponent. So I'll go over PEMDAS, just as a refresher. And then we'll do some examples using both exponents and radicals.

So let's start with a quick review of PEMDAS, which is just our acronym for remembering the order in which we do our different operations. PEMDA stands for Parentheses, Exponents, Multiplying and Dividing, and Adding and Subtracting. So parentheses can include anything like our regular parentheses, but also things like absolute value brackets and radicals. Under exponents, we have our traditional exponents. And then we also have radicals. And again, the reason that we can count radicals under exponents is because you can write a radical as an exponent, and an exponent as a radical. And again, this is what we'll be focusing on today.

With multiplying and dividing, and adding and subtracting, the thing to remember is that multiplying and dividing should be done together from left to right, and adding and subtracting should also be done together, from left to right. So let's do some examples using both radicals and exponents to see how to simplify.

So here's my first example. I've got 3 times 6 minus a 4 divided by negative 2 squared. The parentheses are telling us that this is the part of the expression that we're squaring, so we're not using the exponent for the three that's on the outside, only what's in the parentheses. Thinking of PEMDAS, I'm going to start with my parentheses first and then move onto my exponent and then multiply by the 3.

So in my parentheses, I've got minusing and dividing. So we'll start with dividing. 4 divided by negative 2 is going to give me a negative 2. I'll bring down the rest of my problem in the parentheses. 6 minus negative 2 is going to give me a positive 8. And now that I've finished evaluating what's in my parentheses, I'll bring down my exponent and the 3 that's on the outside.

So again, thinking of order of operations, I first need to do my exponent and then multiply by 3. 8 to the second power, or 8 times 8, is going to give me a 64. Bring down my 3. And 64 times 3 is going to give me 192. So here's my second example. We're going to look at one with a radical in it. In this case, it's a cubed root. It looks like a square root with a 3 on the inside.

So the problem is 10 minus the cubed root of 125 minus 3 times negative 8. So as we've been talking about, we're going to start with our radical. Because that will be evaluated under exponents, and we don't have any parentheses, so we're going to go ahead and start with our radical. The cubed root of 125 means what number multiplied by itself 3 times will give me 125? And that is going to be a 5.

Now that I've finished evaluating my radical, I'm going to bring down the rest of my problem. 10 minus in front. And I've got 10 minus 5, minus 3 times negative 8. So I've got subtracting and multiplying, thinking of PEMDAS, multiplying comes first. 3 times a negative 8 is going to give me negative 24. I'm going to stick it in parentheses, because it's a negative number. Bring down the rest of my problem. And now, I have simply subtracting. So 10 minus 5 minus negative 24, we can do that from left to right. 10 minus 5 will give me 5. And 5 minus a negative 24 is going to give me a positive 29 for my final answer.

So here's my third and final example. I've got a negative 2 to the 4th power, and then in parentheses, 3 minus a negative 1 divided by the square root of 16. If you're feeling pretty confident with exponents and radicals, go ahead and pause and then check back with us and see if you got the right answer. So I'm going to start with my parentheses. I've got 3 minus a negative 1, and that's going to give me a positive 4. I'll bring down the rest of my problem.

After parentheses, we know that we have exponents, but that includes our radicals. So we're going to evaluate the exponent and the radical for our next step. And it doesn't matter which one we do first. Now I've got a negative in front of my 2 to the 4th power. But this is not negative 2 to the 4th power, this is a regular 2 to the 4th with a negative in front. So 2 to the 4th power is 2 times 2 times 2 times 2, which would give me 16. And I'll bring down my negative in front.

And then the square root of 16 is going to give me 4. So now that I've evaluated my exponent and my radical, I'm going to bring down the rest of my problem. I'll divide by 4, divided by 4. And I've got only multiplying and dividing left. And I know that I can just do that together from left to right. So a negative 16 times 4 is going to give me a negative 64. And then if I go ahead and divide that by 4, that's just going to bring me back to negative 16 for my final answer.

So today we talked about how radicals and exponents fit together in the order of operations. We did some examples to help us see how they would be used to simplify examples. The important thing to remember is that radicals, such as the square root, a cubed root or radical with any number in the corner here can be written as an exponent. And thus we evaluated as the E, for exponent, when we're thinking about PEMDAS. So keep using your notes, keep looking at your examples, and keep on practicing. And soon you'll be a pro. Thanks for watching.