Answer
$\sqrt[15]{9}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Convert the given expression, $
\sqrt[5]{\sqrt[3]{9}}
,$ in exponential form by using the definition of rational exponents. Then use the laws of exponents to simplify the exponent. Finally, use the definition of rational exponents again to express the answer in radical form.
$\bf{\text{Solution Details:}}$
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[5]{9^{\frac{1}{3}}}
\\\\=
\left(9^{\frac{1}{3}}\right)^{\frac{1}{5}}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
9^{\frac{1}{3}\cdot\frac{1}{5}}
\\\\=
9^{\frac{1}{15}}
\\\\=
\sqrt[15]{9^{1}}
\\\\=
\sqrt[15]{9}
.\end{array}
