SOLUTION: Simplify ^5 the square root of 243x^18/y^10 by taking roots of the numerator and the denominator and assume that all expressions under radicals represent positive numbers.

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Question 401443: Simplify ^5 the square root of 243x^18/y^10 by taking roots of the numerator and the denominator and assume that all expressions under radicals represent positive numbers.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%285%2C+243x%5E18%2Fy%5E10%29
First, this is called a fifth (or 5th) root. The phrase "square root" has nothing to do with these.

What you are calling a square root is actually the radical symbol. The radical symbol is used for all types of roots. The little number in the upper left corner of the radical is called the index. The index indicates which type of root the radical represents. The index of 5 on your radical tells us that it represents a 5th root. A radical without a visible index has an implied index of 2 and represents a square (or 2nd) root. This is why radicals and square roots can be easily confused.

As the instructions tell you, we will be using a property of radicals, root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29, to split this 5th root of a fraction into a fraction of 5th roots. But first I like to
  1. reduce the fraction.
  2. make the denominator a perfect power of the type of root. (In this case we would make the denominator a perfect power of 5.)

Your fraction is already reduced. And with the way exponents work, any exponent that is a multiply of 5 represents a power of 5. And since the exponent of 10 on y%5E10 is a multiple of 5, the denominator is already a power od 5! So we are ready to split the radical:
root%285%2C+243x%5E18%29%2Froot%285%2C+y%5E10%29
Now we simplify the numerator and denominator radicals. Simplifying 5th roots involves finding factors that are powers of 5. As it happens, 243 is a power of 5! It is 3%5E5. x%5E18 is not a power of 5 since its exponent is not a multiple of 5. But it does have a power of 5 factor: x%5E18+=+x%5E15%2Ax%5E3. And we already know that y%5E10 is a power of 5. Factoring the radicands we get:
root%285%2C+243x%5E15%2Ax%5E3%29%2Froot%285%2C+y%5E10%29
Now we can use another property of radicals, root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29, (from right to left) to split up the radicals so that the power of 5 factors are each in their own radical (and the rest of the factors all go into one additional radical):

We can now simplify the 5th roots of the powers of 5:

%283%2Ax%5E3%2Aroot%285%2C+x%5E3%29%29%2Fy%5E2
And we are finished.

P.S. The part about "all expressions under radicals represent positive numbers" is irrelevant when working with odd-numbered roots (like 5th roots).