If the two numbers are a and b, then the geometric mean between them is given by $\rm GM = \sqrt{ab}$.

Given,

$\rm a = 8$ and $\rm b = \frac{32}{3}$

By substituting the values into the formula, we get,

$\rm or, GM = \sqrt{ab}$

$\rm or, GM =  \sqrt{ 8 \cdot \frac{32}{3} }$

$\rm or, GM =  \sqrt{ \frac{256}{3} }$

$\rm or, GM = \sqrt{ \left ( \frac{16}{\sqrt{3}} \right )^{2} }$

$\rm \therefore GM = \frac{16}{\sqrt{3}}$

Hence, the required geometric mean between the numbers $\rm 8$ and $\rm \frac{32}{3}$ is $\rm \frac{16}{\sqrt{3}}$.