document.write( "Question 656225: Simplify the 5th square root of x^3y^2 10th square root of x^4y^16 \n" ); document.write( "
Algebra.Com's Answer #409591 by jsmallt9(3758)\"\" \"About 
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First of all, there are no such things as \"5th square root\" or \"10th square root\". I'm guessing what you mean is:
\n" ); document.write( "\"root%285%2C+x%5E3y%5E2%29%2Aroot%2810%2C+x%5E4y%5E16%29\"
\n" ); document.write( "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.

\n" ); document.write( "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:
\n" ); document.write( "\"%28x%5E3y%5E2%29%5E%281%2F5%29%2A%28x%5E4y%5E16%29%5E%281%2F10%29\"

\n" ); document.write( "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:
\n" ); document.write( "
\n" ); document.write( "Now we can use the power of a power rule, \"%28a%5En%29%5Em+=+a%5E%28n%2Am%29\":
\n" ); document.write( "
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"x%5E%283%2F5%29%2Ay%5E%282%2F5%29%2Ax%5E%284%2F10%29%2Ay%5E%2816%2F10%29\"
\n" ); document.write( "\"x%5E%283%2F5%29%2Ay%5E%282%2F5%29%2Ax%5E%282%2F5%29%2Ay%5E%288%2F5%29\"
\n" ); document.write( "And finally we can use the \"a%5En%2Aa%5Em+=+a%5E%28n%2Bm%29\" rule to multiply the x's and y's:
\n" ); document.write( "\"x%5E%28%283%2F5%29%2B%282%2F5%29%29%2Ay%5E%28%282%2F5%29%2B%288%2F5%29%29\"
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"x%5E%285%2F5%29%2Ay%5E%2810%2F5%29%29\"
\n" ); document.write( "\"x%5E1%2Ay%5E2\"
\n" ); document.write( "\"xy%5E2\" \n" ); document.write( "
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