To remove the radical from the denominator, multiply both the numerator and the denominator by a value that makes the denominator whole.
If the denominator is cube root(a) multiply the numerator and denominator by cube root(a²).
The numerator becomes the cube root(a²) while the denominator becomes the cube root(a³), which is just a.
So the answer is: cube root(a²) / a.
$$\\\dfrac{1}{\sqrt[3]{a}}\\\\\\
=\dfrac{1}{a^{1/3}}\\\\\\
=\dfrac{1}{a^{1/3}}\times \dfrac{a^{2/3}}{a^{2/3}}\\\\\\
=\dfrac{a^{2/3}}{a^{1/3+2/3}}\\\\\\
=\dfrac{a^{2/3}}{a^{1}}\\\\\\
=\dfrac{\sqrt[3]{a^2}}{a}\\\\\\
$Which is exactly what geno said in the first place! $$$