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Simplifying higher-index root expressions

How to rewrite a radical with variables in it as an exponent and then simplify it using exponent properties. In this example, we simplify ∜(5a⁴b¹²). Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Rewrite the radical expression using rational exponents and simplify. So here we have the fourth root of 5 a to the fourth b to the twelfth power. The key thing to realize here is the fourth root of something is same thing as something to the one fourth power. Or in particular, or in general, the nth root something is the same thing as that something to the 1 over n power. So we can just apply that over here: the fourth root of all of this is equal to 5 a to the 4th, b to the 12th power, all of that to the one fourth power. And then we also know if we take the product of things, and then raise them to some exponent, that's the same thing as raising each of the terms in the product to the exponent first, or each of the things that we're taking the product of to that exponent, and then multiplying. So let's do that. So 5, so this is the same thing as 5 to the 1/4 power times a to the fourth to the 1/4 power times b to the 12th to the 1/4 power. Now 5 to the 1/4, I don't know what that is, so I'll just keep that as the cube root, well, we could leave it as 5 to the 1/4, and that's not not simplified. or we can just rewrite it again as the fourth root of 5. a to the fourth to the 1/4 power: if your raise something to a power and then another power, and raise that to another power, that's equivalent of raising a to the four times 1/4 power. So let me just write that down. This is, so times a to four times 1/4 power. And then finally, this right over here, using the same exact exponent property, this is b to the 12th times 1/4 power. So all of this simplifies to, and I'll change the order here, so you have the fourth root of 5, and then you have a to the fourth times 1/4 power so that's just, this simplifies to a to the first power which is really just the same thing as a. So that's just a. And then we have b to the 12 times 1/4 power. Well, 12 times 1/4 is just three. So that's b to the third power. So it's a, b to the third power times the fourth root of 5.