document.write( "Question 354941: ^5sqrt(96x^5) \n" ); document.write( "
Algebra.Com's Answer #253955 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Simplifying radicals involves:
  • Eliminating any fractions within a radical
  • Eliminating any radicals in the denominator of a fraction
  • \"Reducing\" the radical by factoring out perfect powers of the type of root.

\n" ); document.write( "Since your expression (pronounced \"the 5th root of 96 x to the 5th power\")
\n" ); document.write( "\"root%285%2C+96x%5E5%29\"
\n" ); document.write( "has no fractions, we can skip the first two parts of simplifying. Now we just look for perfect power factors. Since your expression is a 5th root, then we look for factors which are perfect powers of 5. Obviously \"x%5E5\" is a power of 5. But we also look for perfect powers of 5 in the 96. Since \"2%5E5+=+32\" and since 96 = 32*3, there is another power of 5 factor, 32, in your expression. Rewriting your expression with its radicand factored we get:
\n" ); document.write( "\"root%285%2C+32%2Ax%5E5%2A3%29\"
\n" ); document.write( "(Since multiplication is Commutative, the order of the factors is not important. I like to order the factors with the perfect power factors first and other factors, if any, at the end.)
\n" ); document.write( "Now we can use a basic property of all radicals, \"root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29\", to separate each factor into its own personal radical:
\n" ); document.write( "\"root%285%2C+32%29%2Aroot%285%2C+x%5E5%29%2Aroot%286%2C+3%29\"
\n" ); document.write( "The 5th roots of the power of 5 factors are easy to find:
\n" ); document.write( "\"2%2Ax%2Aroot%285%2C+3%29\"
\n" ); document.write( "or
\n" ); document.write( "\"2x%2Aroot%285%2C+3%29\" \n" ); document.write( "
\n" );