Answer
$a)$ $4st^{4}$
$b)$ $\dfrac{4x}{y}$
Work Step by Step
$a)$ $\dfrac{(8s^{3}t^{3})^{2/3}}{(s^{4}t^{-8})^{1/4}}$
Evaluate the powers in the numerator and the denominator:
$\dfrac{(8s^{3}t^{3})^{2/3}}{(s^{4}t^{-8})^{1/4}}=\dfrac{(8^{2/3})s^{2}t^{2}}{st^{-2}}=\dfrac{(\sqrt[3]{8^{2}})s^{2}t^{2}}{st^{-2}}=\dfrac{4s^{2}t^{2}}{st^{-2}}=...$
Evaluate the division:
$...=4s^{2-1}t^{2-(-2)}=4st^{4}$
$b)$ $\dfrac{(32x^{5}y^{-3/2})^{2/5}}{(x^{5/3}y^{2/3})^{3/5}}$
Evaluate the powers in the numerator and the denominator:
$\dfrac{(32x^{5}y^{-3/2})^{2/5}}{(x^{5/3}y^{2/3})^{3/5}}=\dfrac{(32^{2/5})x^{2}y^{-3/5}}{xy^{2/5}}=\dfrac{(\sqrt[5]{32^{2}})x^{2}y^{-3/5}}{xy^{2/5}}=...$
$...=\dfrac{4x^{2}y^{-3/5}}{xy^{2/5}}=...$
Evaluate the division and simplify:
$...=4x^{2-1}y^{-3/5-2/5}=4xy^{-5/5}=4xy^{-1}=\dfrac{4x}{y}$
