The question is simplify the expression $\left(\dfrac{a^2}{27}\right)^{1/3}\left(\dfrac{64}a\right)^{2/3}$
1: Multiply on both sides equals $\dfrac{a^{2/3}}{27^{1/3}}\cdot \dfrac{64^{2/3}}{a^{2/3}}$
Does this give me $\dfrac{a^{2/3}}{3} \cdot \dfrac{16}{a^{2/3}}$ ?
Assuming that $a\neq 0$; then
$$\Big(\frac{64}{a}\Big)^{\frac{2}{3}}=\Big(\frac{4^6}{a^2}\Big)^{\frac{1}{3}};$$
then $$\Big(\frac{a^2}{3^3}\Big)^{\frac{1}{3}}\Big(\frac{16^3}{a^2}\Big)^{\frac{1}{3}}=\frac{16}{3}.$$
So your answer is correct. Clearly if $a=0$ this expression is not define.
Notice, $$\left(\frac{a^2}{27}\right)^{1/3}\cdot\left(\frac{64}{a}\right)^{2/3}$$ $$=\left(\frac{(a)^{2/3}}{(27)^{1/3}}\right)\cdot\left(\frac{(64)^{2/3}}{(a)^{2/3}}\right)$$ $$=\left(\frac{1}{3}\right)\cdot\left(4^2\right)=\color{blue}{\frac{16}{3}}$$
