Question

Find the value of $(1000\pi {)}^{2/3}\times (512\pi {)}^{4/3}$

Answer 1

Observe $1000=2^3\times 5^3$ and $512=2^9$. Hence
$(1000\pi {)}^{2/3}\times (512\pi {)}^{4/3}=(2^3{5}^3\pi {)}^{2/3}\times (2^9\pi {)}^{4/3}$
$=(2^3{)}^{2/3}\times (5^3{)}^{2/3}\times (\pi {)}^{2/3}\times (2^9{)}^{4/3}\times (\pi {)}^{4/3}$
$=2^2\times 5^2\times 2^{12}\times (\pi {)}^{2/3}\times (\pi {)}^{4/3}$
$=2^{14}{5}^2(\pi {)}^{\left(2/3\right)+\left(4/3\right)}=409600{\pi }^2$.