SOLUTION: Simplify the 5th square root of x^3y^2 10th square root of x^4y^16

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Question 656225: Simplify the 5th square root of x^3y^2 10th square root of x^4y^16
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, there are no such things as "5th square root" or "10th square root". I'm guessing what you mean is:
root%285%2C+x%5E3y%5E2%29%2Aroot%2810%2C+x%5E4y%5E16%29
If this is correct then I think working with these expressions will be easier for you if you learn what they are. The first radical represents a "5th root" not a "5th square root" and the second one is a "10th root" not a "10th square root". The absence of the word "square" is important. The first one represents whatever expression you could raise to the 5th power and get x%5E3y%5E2 as a result. And the second one represents whatever expression you could raise to the 10th power and get x%5E4y%5E16 as a result.

I think the easiest way to simplify this expression is to use fractional exponents. For 5th roots the exponent is 1/5 and for 10th roots the exponent is 1/10. So we can rewrite the expression using these exponents instead of the radicals:
%28x%5E3y%5E2%29%5E%281%2F5%29%2A%28x%5E4y%5E16%29%5E%281%2F10%29

Now we can start using our rules for exponents to simplify. First we will raise to 1/5 and 1/10 powers using the %28a%2Ab%29%5En+=+a%5En%2Ab%5En rule:

Now we can use the power of a power rule, %28a%5En%29%5Em+=+a%5E%28n%2Am%29:

which simplifies as follows:
x%5E%283%2F5%29%2Ay%5E%282%2F5%29%2Ax%5E%284%2F10%29%2Ay%5E%2816%2F10%29
x%5E%283%2F5%29%2Ay%5E%282%2F5%29%2Ax%5E%282%2F5%29%2Ay%5E%288%2F5%29
And finally we can use the a%5En%2Aa%5Em+=+a%5E%28n%2Bm%29 rule to multiply the x's and y's:
x%5E%28%283%2F5%29%2B%282%2F5%29%29%2Ay%5E%28%282%2F5%29%2B%288%2F5%29%29
which simplifies as follows:
x%5E%285%2F5%29%2Ay%5E%2810%2F5%29%29
x%5E1%2Ay%5E2
xy%5E2