# Exponents: Rational exponents

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##### Examples
###### Lessons
1. prove: $a^{3 \over 8} = {^8}\sqrt{a^3}$
1. Simplifying Expressions Using: ${^n}\sqrt{x}=x^{\frac{1}{n}}$
Simplify the following expressions if possible.
1. $64^{\frac{1}{3}}$
$16^{\frac{1}{4}}$
2. $(-16)^{\frac{1}{4}}$
$(-32)^{\frac{1}{5}}$
2. evaluate:
1. $(25)^{1 \over 2}$
2. $(-4)^{1 \over 2}$
3. $(10)^{3 \over 8}$
4. $(8)^{5 \over 3}$
5. $(-{243 \over 32})^{-{2 \over 5}}$
3. Simplifying Expressions Using: $x^{-\frac{1}{n}}=\frac{1}{x^{\frac{1}{n}}}=\frac{1}{{^n}\sqrt{x}}$
Simplify the following expressions.
1. $27^{-\frac{1}{3}}$
2. $\frac{1}{{^6}\sqrt{x}}$
3. $(64x^8)^{-\frac{1}{2}}$
4. Simplifying Expressions Using: $x^{\frac{m}{n}}={^n}\sqrt{x^m}$
Simplify the following expressions if possible.
1. ${^2}\sqrt{x^6}$
2. $25^{\frac{3}{2}}$
3. $(-125)^{-\frac{2}{3}}$
4. $\sqrt{36x^{16}y^{24}}$
5. ${^3}\sqrt{-216a^9b^{24}c^{117}}$
###### Topic Notes
${^n}\sqrt{x}=x^{\frac{1}{n}}$
$x^{-\frac{1}{n}}=\frac{1}{x^{\frac{1}{n}}}=\frac{1}{{^n}\sqrt{x}}$
$x^{\frac{m}{n}}={^n}\sqrt{x^m}$