Answer
$\sqrt[12]{a}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the definition of rational exponents and the laws of exponents to simplify the given expression, $
\sqrt[6]{\sqrt{a}}
.$
$\bf{\text{Solution Details:}}$
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[6]{\sqrt{a}}
\\\\=
\sqrt[6]{a^{1/2}}
\\\\=
\left( a^{1/2} \right)^{1/6}
.\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}
\left( a^{1/2} \right)^{1/6}
\\\\=
a^{\frac{1}{2}\cdot\frac{1}{6}}
\\\\=
a^{\frac{1}{12}}
\\\\=
\sqrt[12]{a}
.\end{array}
