Answer
$a)$ $\dfrac{4r^{5/2}}{s^{7}}$
$b)$ $\dfrac{4a^{4}c^{6}}{b^{12}}$
Work Step by Step
$a)$ $\dfrac{8r^{1/2}s^{-3}}{2r^{-2}s^{4}}$
Evaluate the division:
$\dfrac{8r^{1/2}s^{-3}}{2r^{-2}s^{4}}=\dfrac{8}{2}r^{1/2+2}s^{-3-4}=4r^{5/2}s^{-7}=...$
Take $s^{-7}$ to the denominator to change the sign of its exponent:
$...=\dfrac{4r^{5/2}}{s^{7}}$
$b)$ $\Big(\dfrac{ab^{2}c^{-3}}{2a^{3}b^{-4}}\Big)^{-2}$
Evaluate the power:
$\Big(\dfrac{ab^{2}c^{-3}}{2a^{3}b^{-4}}\Big)^{-2}=\dfrac{a^{-2}b^{-4}c^{6}}{2^{-2}a^{-6}b^{8}}=...$
Evaluate the division and simplify:
$...=\dfrac{1}{2^{-2}}a^{-2+6}b^{-4-8}c^{6}=(2^{2})a^{4}b^{-12}c^{6}=\dfrac{4a^{4}c^{6}}{b^{12}}$
