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# Simplifying higher-index root expressions

CCSS.Math:

## Video transcript

rewrite the radical expression using rational exponents and to simplify so here we have the fourth root of five a to the fourth B to the 12th power and the key thing to realize here is the fourth root of something is the same thing as that something to the 1/4 power or in particular the or in general the nth root of something is the same thing as that something to the one over N power so we can just apply that over here the fourth root of all of this is equal to five a to the fourth B to the twelfth power all of that to the 1/4 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 and then multiplying so let's do that so five so this is the same thing as five to the one fourth power times a to the fourth to the one fourth power times B to the 12th to the one fourth power now five to the one fourth I don't know what that is so I'll just keep that as the cube root well we could leave it as five to the one fourth and that's not not simplified or we could just rewrite it again as the fourth root fourth root of five a to the fourth to the one fourth power if you raise something to a power and then another power and raise that to another power that's the equivalent of raising the a to the four times one fourth power so let me just write that down this is so times a to the four times one fourth power and then finally this right over here using the same exact exponent property this is B to the 12th times one fourth power and so all of this simplifies to and I'll change the order here so you have the Q the fourth root the fourth root of five and then you have a to the fourth times one fourth 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 twelve times 1/12 times one fourth our well 12 times one fourth 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 five