Answer
$a)$ $\dfrac{2y^{4/3}}{x^{2}}$
$b)$ $\dfrac{t^{1/4}}{2s^{1/2}}$
Work Step by Step
$a)$ $\Big(\dfrac{x^{8}y^{-4}}{16y^{4/3}}\Big)^{-1/4}$
Invert the fraction to change the sign of its exponent:
$\Big(\dfrac{x^{8}y^{-4}}{16y^{4/3}}\Big)^{-1/4}=\Big(\dfrac{16y^{4/3}}{x^{8}y^{-4}}\Big)^{1/4}=...$
Evaluate the power:
$...=\dfrac{(16^{1/4})y^{1/3}}{x^{2}y^{-1}}=\dfrac{(\sqrt[4]{16})y^{1/3}}{x^{2}y^{-1}}=\dfrac{2y^{1/3}}{x^{2}y^{-1}}=...$
Evaluate the division and simplify if possible:
$...=\dfrac{2y^{1/3-(-1)}}{x^{2}}=\dfrac{2y^{4/3}}{x^{2}}$
$b)$ $\Big(\dfrac{4s^{3}t^{4}}{s^{2}t^{9/2}}\Big)^{-1/2}$
Invert the fraction to change the sign of its exponent:
$\Big(\dfrac{4s^{3}t^{4}}{s^{2}t^{9/2}}\Big)^{-1/2}=\Big(\dfrac{s^{2}t^{9/2}}{4s^{3}t^{4}}\Big)^{1/2}=...$
Evaluate the power:
$...=\dfrac{st^{9/4}}{(\sqrt{4})s^{3/2}t^{2}}=\dfrac{st^{9/4}}{2s^{3/2}t^{2}}=...$
Evaluate the division and simplify if possible:
$...=\dfrac{s^{1-3/2}t^{9/4-2}}{2}=\dfrac{s^{-1/2}t^{1/4}}{2}=\dfrac{t^{1/4}}{2s^{1/2}}$
