Question 354941
Simplifying radicals involves:<ul><li>Eliminating any fractions within a radical</li><li>Eliminating any radicals in the denominator of a fraction</li><li>"Reducing" the radical by factoring out perfect powers of the type of root.</li></ul>
Since your expression (pronounced "the 5th root of 96 x to the 5th power")
{{{root(5, 96x^5)}}}
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^5}}} is a power of 5. But we also look for perfect powers of 5 in the 96. Since {{{2^5 = 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:
{{{root(5, 32*x^5*3)}}}
(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.)
Now we can use a basic property of all radicals, {{{root(a, p*q) = root(a, p)*root(a, q)}}}, to separate each factor into its own personal radical:
{{{root(5, 32)*root(5, x^5)*root(6, 3)}}}
The 5th roots of the power of 5 factors are easy to find:
{{{2*x*root(5, 3)}}}
or
{{{2x*root(5, 3)}}}