Answer
$a)$ $\dfrac{\sqrt{3st}}{3t}$
$b)$ $\dfrac{a\sqrt[3]{b^{2}}}{b}$
$c)$ $\dfrac{\sqrt[5]{c^{2}}}{c}$
Work Step by Step
$a)$ $\sqrt{\dfrac{s}{3t}}$
Rewrite as $\dfrac{\sqrt{s}}{\sqrt{3t}}$:
$\sqrt{\dfrac{s}{3t}}=\dfrac{\sqrt{s}}{\sqrt{3t}}=...$
Multiply the numerator and the denominator by $\sqrt{3t}$ and simplify:
$...=\dfrac{\sqrt{s}}{\sqrt{3t}}\cdot\dfrac{\sqrt{3t}}{\sqrt{3t}}=\dfrac{\sqrt{3st}}{\sqrt{(3t)^{2}}}=\dfrac{\sqrt{3st}}{3t}$
$b)$ $\dfrac{a}{\sqrt[6]{b^{2}}}$
Multiply the numerator and the denominator by $\sqrt[6]{b^{4}}$ and simplify:
$\dfrac{a}{\sqrt[6]{b^{2}}}=\dfrac{a}{\sqrt[6]{b^{2}}}\cdot\dfrac{\sqrt[6]{b^{4}}}{\sqrt[6]{b^{4}}}=\dfrac{a\sqrt[6]{b^{4}}}{\sqrt[6]{b^{6}}}=\dfrac{a\sqrt[6]{b^{4}}}{b}=\dfrac{ab^{4/6}}{b}=\dfrac{ab^{2/3}}{b}=\dfrac{a\sqrt[3]{b^{2}}}{b}$
$c)$ $\dfrac{1}{c^{3/5}}$
Rewrite as $\dfrac{1}{\sqrt[5]{c^{3}}}$:
$\dfrac{1}{c^{3/5}}=\dfrac{1}{\sqrt[5]{c^{3}}}=...$
Multiply the numerator and the denominator by $\sqrt[5]{c^{2}}$ and simplify:
$...=\dfrac{1}{\sqrt[5]{c^{3}}}\cdot\dfrac{\sqrt[5]{c^{2}}}{\sqrt[5]{c^{2}}}=\dfrac{\sqrt[5]{c^{2}}}{\sqrt[5]{c^{5}}}=\dfrac{\sqrt[5]{c^{2}}}{c}$
