Answer
$\dfrac{\sqrt[3]{4}}{2}$
Work Step by Step
Using $\sqrt[n]{x^m}=x^{\frac{m}{n}},$ the given expression, $
\dfrac{\sqrt[6]{4}}{\sqrt[3]{4}}
,$ is equivalent to
\begin{align*}
&
\dfrac{\sqrt[6]{2^2}}{\sqrt[3]{4}}
\\\\&=
\dfrac{2^{\frac{2}{6}}}{2^{\frac{2}{3}}}
.\end{align*}
Using the laws of exponents, the expression above is equivalent to
\begin{align*}
&
2^{\frac{2}{6}-\frac{2}{3}}
&\left( \text{use }\dfrac{a^x}{a^y}=a^{x-y} \right)
\\\\&=
2^{\frac{2}{6}-\frac{4}{6}}
\\\\&=
2^{-\frac{2}{6}}
\\\\&=
2^{-\frac{1}{3}}
\\\\&=
\dfrac{1}{2^{\frac{1}{3}}}
&\left( \text{use }a^{-x}=\dfrac{1}{a^x} \right)
\\\\&=
\dfrac{1}{\sqrt[3]{2^1}}
\\\\&=
\dfrac{1}{\sqrt[3]{2}}
\\\\&=
\dfrac{1}{\sqrt[3]{2}}\cdot\dfrac{\sqrt[3]{2^2}}{\sqrt[3]{2^2}}
&\left( \text{rationalize denominator} \right)
\\\\&=
\dfrac{\sqrt[3]{2^2}}{\sqrt[3]{2^3}}
\\\\&=
\dfrac{\sqrt[3]{4}}{2}
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{\sqrt[3]{4}}{2}
.$
