Answer
a. $ \sqrt {3 f^3}$
b. $ \frac{1}{x^2}$
c. $ \frac{1}{\sqrt 2 \times \sqrt a}$
d. $\frac{1}{\sqrt {8m}}$
Work Step by Step
We can use the following properties to simplify a radical expression:
1. $\sqrt {a^{(-b)}} = \frac{1}{\sqrt {a^{b}}}$
2. $\frac{1}{\sqrt {a^{-b}}} = {\sqrt {a^{b}}}$
a. $\frac{\sqrt 3}{\sqrt {f^{-3}}}$
$= \sqrt 3 \times \sqrt {f^3}$
$= \sqrt {3 \times f^3}$
b. $\frac{\sqrt {x^{(-3)}}}{\sqrt x}$
$= \frac{1}{\sqrt x \times \sqrt {x^3}}$
$= \frac{1}{\sqrt {x \times x^3}}$
$= \frac{1}{\sqrt{x^4}}$
$= \frac{1}{x^2}$
c. $\frac{\sqrt {5a^{-2}}}{\sqrt {10a^{-1}}}$
$= \frac{\sqrt {5a^1}}{\sqrt {10a^2}}$
$= \frac{\sqrt 5 \times \sqrt a}{\sqrt 5 \times \sqrt 2 \times a}$
$= \frac{1}{\sqrt 2 \times \sqrt a}$
d. $\frac{\sqrt {(2m)^{-3}}}{ {m^{-1}}}$
$= \frac{m}{\sqrt {(2m)^3}}$
$= \frac{m}{\sqrt {2^3} \times m \times \sqrt m}$
$=\frac{1}{\sqrt {8m}}$
