Answer
$a)$ $\dfrac{x}{ab^{10}y^{10/3}}$
$b)$ $4s^{3/2}t^{9/2}$
Work Step by Step
$a)$ $\Big(\dfrac{a^{1/6}b^{-3}}{x^{-1}y}\Big)^{3}\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)$
First, evaluate the power:
$\Big(\dfrac{a^{1/6}b^{-3}}{x^{-1}y}\Big)^{3}\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)=\Big(\dfrac{a^{1/2}b^{-9}}{x^{-3}y^{3}}\Big)\Big(\dfrac{x^{-2}b^{-1}}{a^{3/2}y^{1/3}}\Big)=...$
Evaluate the product:
$...=\dfrac{a^{1/2}b^{-10}x^{-2}}{a^{3/2}x^{-3}y^{10/3}}=...$
Finally, evaluate the division and simplify if possible:
$...=\dfrac{a^{1/2-3/2}b^{-10}x^{-2+3}}{y^{10/3}}=\dfrac{a^{-1}b^{-10}x}{y^{10/3}}=\dfrac{x}{ab^{10}y^{10/3}}$
$b)$ $\dfrac{(9st)^{3/2}}{(27s^{3}t^{-4})^{2/3}}\Big(\dfrac{3s^{-2}}{4t^{1/3}}\Big)^{-1}$
Evaluate the powers in the numerator and the denominator of the first fraction and invert the second fraction to change the sign of its exponent:
$\dfrac{(9st)^{3/2}}{(27s^{3}t^{-4})^{2/3}}\Big(\dfrac{3s^{-2}}{4t^{1/3}}\Big)^{-1}=\Big[\dfrac{(\sqrt{9^{3}})s^{3/2}t^{3/2}}{(\sqrt[3]{27^{2}})s^{2}t^{-8/3}}\Big]\Big(\dfrac{4t^{1/3}}{3s^{-2}}\Big)=...$
$...=\Big(\dfrac{27s^{3/2}t^{3/2}}{9s^{2}t^{-8/3}}\Big)\Big(\dfrac{4t^{1/3}}{3s^{-2}}\Big)=...$
Evaluate the product of fractions:
$...=\dfrac{108s^{3/2}t^{3/2+1/3}}{27s^{2-2}t^{-8/3}}=\dfrac{108s^{3/2}t^{11/6}}{27t^{-8/3}}=...$
Evaluate the division and simplify:
$...=\Big(\dfrac{108}{27}\Big)s^{3/2}t^{11/6+8/3}=4s^{3/2}t^{9/2}$
