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Simplifying Radicals using Rational Exponents - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When simplifying roots that are either greater than four or have a term raised to a large number, we rewrite the problem using rational exponents. Remember that every root can be written as a fraction, with the denominator indicating the root's power. When simplifying radicals, since a power to a power multiplies the exponents, the problem is simplified by multiplying together all the exponents.

Every once in a while we're asked to simplify radicals where we actually don't know numerically what the things we're looking at are, so what I have behind me is two ways of writing the exact same thing. We have the sixth root of 5 to the twelve or the six root of 5 out of the 12. Remember we were taking power to power and multiplying so these are actually exactly equivalent statements.
The problem is, just trying to evaluate these on a calculator okay? I don't know what 5 to the twelfth is, so I sure don't know what the sixth root of 5 to twelfth is. Similarly, I don't know what the sixth root of 5 is so then I don't know what the sixth root of 5 is to the twelfth, so we're trying to need figure out a way to somehow deal with this so we can actually simplify without a calculator okay? And what we can do is rewrite these using exponents okay? So what we have here is a root and if we use it rational exponents oops I hope if I write the right number down we have 5 and then we have a power which is going to be power over root so we have the twelfth power over the sixth root so what we end up having is 12, 6 which we know is 2 this ends up giving us 5 squared which we can simplify to 25.
So whenever we're dealing with really big powers or roots you can always look at it and think about if there's a way to simplify these exponents up okay? In this case we just rewrote as a exponential fraction simplified up quite easily and we're able to solve it.

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