# Exponents: Rational exponents

### Exponents: Rational exponents

#### Lessons

${^n}\sqrt{x}=x^{\frac{1}{n}}$
$x^{-\frac{1}{n}}=\frac{1}{x^{\frac{1}{n}}}=\frac{1}{{^n}\sqrt{x}}$
$x^{\frac{m}{n}}={^n}\sqrt{x^m}$
• 1.
prove: $a^{3 \over 8} = {^8}\sqrt{a^3}$

• 2.
Simplifying Expressions Using: ${^n}\sqrt{x}=x^{\frac{1}{n}}$
Simplify the following expressions if possible.
a)
$64^{\frac{1}{3}}$
$16^{\frac{1}{4}}$

b)
$(-16)^{\frac{1}{4}}$
$(-32)^{\frac{1}{5}}$

• 3.
evaluate:
a)
$(8)^{5 \over 3}$

b)
$(-{243 \over 32})^{-{2 \over 5}}$

• 4.
Simplifying Expressions Using: $x^{-\frac{1}{n}}=\frac{1}{x^{\frac{1}{n}}}=\frac{1}{{^n}\sqrt{x}}$
Simplify the following expressions.
a)
$27^{-\frac{1}{3}}$

b)
$\frac{1}{{^6}\sqrt{x}}$

c)
$(64x^8)^{-\frac{1}{2}}$

• 5.
Simplifying Expressions Using: $x^{\frac{m}{n}}={^n}\sqrt{x^m}$
Simplify the following expressions if possible.
a)
${^2}\sqrt{x^6}$

b)
$25^{\frac{3}{2}}$

c)
$(-125)^{-\frac{2}{3}}$

d)
$\sqrt{36x^{16}y^{24}}$

e)
${^3}\sqrt{-216a^9b^{24}c^{117}}$