Answer
$a)$ $\dfrac{\sqrt{5x}}{5x}$
$b)$ $\dfrac{\sqrt{5x}}{5}$
$c)$ $\dfrac{\sqrt[5]{x^{2}}}{x}$
Work Step by Step
$a)$ $\dfrac{1}{\sqrt{5x}}$
Multiply the numerator and the denominator by $\sqrt{5x}$ and simplify:
$\dfrac{1}{\sqrt{5x}}=\dfrac{1}{\sqrt{5x}}\cdot\dfrac{\sqrt{5x}}{\sqrt{5x}}=\dfrac{\sqrt{5x}}{\sqrt{(5x)^{2}}}=\dfrac{\sqrt{5x}}{5x}$
$b)$ $\sqrt{\dfrac{x}{5}}$
Rewrite as $\dfrac{\sqrt{x}}{\sqrt{5}}$:
$\sqrt{\dfrac{x}{5}}=\dfrac{\sqrt{x}}{\sqrt{5}}=...$
Multiply the numerator and the denominator by $\sqrt{5}$ and simplify:
$...=\dfrac{\sqrt{x}}{\sqrt{5}}\cdot\dfrac{\sqrt{5}}{\sqrt{5}}=\dfrac{\sqrt{5x}}{\sqrt{5^{2}}}=\dfrac{\sqrt{5x}}{5}$
$c)$ $\sqrt[5]{\dfrac{1}{x^{3}}}$
Rewrite as $\dfrac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}$ and simplify:
$\sqrt[5]{\dfrac{1}{x^{3}}}=\dfrac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}=\dfrac{1}{\sqrt[5]{x^{3}}}=...$
Multiply the numerator and the denominator by $\sqrt[5]{x^{2}}$ and simplify again:
$...=\dfrac{1}{\sqrt[5]{x^{3}}}\cdot\dfrac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{2}}}=\dfrac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{5}}}=\dfrac{\sqrt[5]{x^{2}}}{x}$
