document.write( "Question 426357: n-th root of [32/{2^(5+n)}]\r
\n" ); document.write( "\n" ); document.write( "I am having difficulties with this problem. Please guide me through it. Thank you. \n" ); document.write( "

Algebra.Com's Answer #296793 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"root%28n%2C+32%2F2%5E%285%2Bn%29%29\"$
\n" ); document.write( "The keys to this problem are:

Recognizing (or figuring out) that $\"2%5E5+=+32\"$
Knowing that the rules for exponents work in both directions.

\n" ); document.write( "If we were to multiply $\"2%5E5%2A2%5En\"$ the rule for exponents would say to add the exponents. So
\n" ); document.write( " $\"2%5E5%2A2%5En+=+2%5E%285%2Bn%29\"$
\n" ); document.write( "It works backwards, too. Your denominator is the right side of the above equation. So we can replace your denominator with the left side:
\n" ); document.write( " $\"root%28n%2C+32%2F%282%5E5%2A2%5En%29%29\"$
\n" ); document.write( "Since $\"2%5E5+=+32\"$ they cancel each other out in the fraction:
\n" ); document.write( " $\"root%28n%2C+cross%2832%29%2F%28cross%282%5E5%29%2A2%5En%29%29\"$
\n" ); document.write( "leaving:
\n" ); document.write( " $\"root%28n%2C+1%2F2%5En%29\"$
\n" ); document.write( "Now we can use a property of radicals, $\"root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29\"$ , to split this radical:
\n" ); document.write( " $\"root%28n%2C+1%29%2Froot%28n%2C+2%5En%29\"$
\n" ); document.write( "And both of these nth roots simplify:
\n" ); document.write( "1/2 \n" ); document.write( "

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