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Simplifying Radicals using Rational Exponents - Problem 2

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!

Simplifying an expression with a ton of square roots and powers. So this expression behind me is pretty ugly, we have a lot going on and what we want to do is simplify this somehow.

The first step in simplifying this is to always rewrite it as rational exponents. So just starting at the inside, I start with x to the twelfth, I'm taking the fourth root of that, so I want to take fourth root write it as a exponent, power over root this is just going to be to the 1/4, so that takes care of the twelfth and the fourth root.

Going down to 9, the next thing we're doing is the cube root, rewriting the cube root as a exponent is 1/3, going down the line again, down to the square root, square root is just going to be to the 1/2 and the last thing we have is to the sixth power which stays just to sixth. I've left off all my parenthesis over here that's perfectly fine because I'm really just concerned with the power of all this.

Now what we're dealing is we have a bunch of powers to power, when we take power to power and multiply, so what we really want to do is just multiply all this out. We could distribute it all through, but we can also just cancel things that simplify to make our life a little bit easier, so we have a 12 in the numerator and a 4 and 3 in the denominator so those all cancel out leaving us with just a 1.

We also have a 6 and a 1/2, so that just cancels to 3, and so what we really have is everything is cancelled except for a 3 in the numerator leaving us with x³. So we were able to take a fairly ugly expression and just by rewriting using rational exponents, it will simplify quite easily.

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