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Simplifying Radicals using Rational Exponents - Concept
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.