document.write( "Question 242613: root:5:(64x^(8)y^(12))
\n" ); document.write( "How do I pull all perfect 5th rotts out? \n" ); document.write( "

Algebra.Com's Answer #177603 by jsmallt9(3758) $\"\"$ $\"About$

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$\"root%285%2C+64x%5E8y%5E12%29\"$

\n" ); document.write( "To simplify this we need to find factors that are perfect powers of 5, if any.

\n" ); document.write( "For the 64 we look for factors of 64 that are perfect powers of 5, if any. Since $\"32+=+2%5E5\"$ and 64 = 32*2, we do have a perfect power of 5 factor in 64.

\n" ); document.write( "For the variables we can factor out perfect powers of 5 as long as the exponent is greater than or equal to 5.

\n" ); document.write( " $\"root%285%2C+64x%5E8y%5E12%29\"$
\n" ); document.write( " $\"root%285%2C+32%2A2%2Ax%5E5%2Ax%5E3%2Ay%5E5%2Ay%5E7%29\"$ (There's another power of 5 in $\"y%5E7\"$ )
\n" ); document.write( " $\"root%285%2C+32%2A2%2Ax%5E5%2Ax%5E3%2Ay%5E5%2Ay%5E5%2Ay%5E2%29\"$
\n" ); document.write( "Now we can use the property of radicals, $\"root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29\"$ , to split up all these factors into separate radicals:
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\n" ); document.write( "Next, for each factor that is a perfect power of 5, we substitute in its value:
\n" ); document.write( " $\"2%2Aroot%285%2C+2%29%2Ax%2Aroot%285%2C+x%5E3%29%2Ay%2Ay%2Aroot%285%2C+y%5E2%29\"$
\n" ); document.write( "Now we use the Commutative and Associative Properties to rearrange and regroup the factors. We want the non-radical factors grouped together and the radical factors grouped together. (The convention is to put the radical at the end):
\n" ); document.write( " $\"%282%2Ax%2Ay%2Ay%29%2A%28root%285%2C+2%29%2Aroot%285%2C+x%5E3%29%2Aroot%285%2C+y%5E2%29%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"2xy%5E2%5Croot%285%2C+2x%5E3y%5E2%29\"$ \n" ); document.write( "

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