# Radicals and Expressions with Rational Exponents

1.  $\rhd$ Introduction to rational exponents  (exponents are unit fractions such as $\frac{1}{2}$ and $\frac{1}{3}$ (4:59)
–[Example 1] $4^3 = 64$
–[Example 2] $4^{-3} = \frac{1}{64}$
–[Example 3] $4^{\frac{1}{2}} = \sqrt{4} = 2$
–[Example 4] $8^{\frac{1}{3}} = \sqrt[3]{8} 2$ as $2^3 = 8$
–[Example 5] $32^{\frac{1}{5}} = 2$ as $2^5 = 32$
2.  $\rhd$ Radicals and Exponents (8:45)
–[Example 1] $\sqrt{9} = \sqrt[2]{9} = 3$
–[Example 2] $\sqrt[3]{27} = 3$
–[Example 3] $\sqrt[4]{16} = 2$
–[Example 4] $\sqrt[5]{96} = \sqrt[5]{2^5 \cdot 3} = 2 \cdot \sqrt[5]{3}$
–[Example 5] $\sqrt[6]{64 \cdot x^8} = 2x\sqrt[6]{x^2} = 2 x\cdot \sqrt[3]{x} = 2x^{\frac{4}{3}}$
3.  $\star$ Simplify the exponential expression  $(-128)^{\frac{1}{7}}$
4. $\rhd$ Evaluating rational exponents which are negative unit fractions (3:02)
–[Example 1] $9^{-\frac{1}{2}} = \frac{1}{3}$
–[Example 2] $-27^{-\frac{1}{3}} = -\frac{1}{3}$
5.  $\rhd$ Evaluating rational expressions (5:53)
–[Example 1] $(64)^{\frac{2}{3}} = (4^3)^{\frac{2}{3}} = 4^2 = 16$
–[Example 2] $(\frac{8}{27})^{-2}{3} = \frac{9}{4}$
6.  $\rhd$ Evaluating rational expressions with rational bases (3:24) $(\frac{25}{9})^{\frac{1}{2}} = \frac{5}{3}$ and $(\frac{81}{256})^{-\frac{1}{4}} = \frac{4}{3}$
7.  $\rhd$ Converting from radical to fractional exponents (4:01) If $3^a = \sqrt[3]{\frac{3}{2}}$, then $a = \frac{2}{5}$
8.  $\star$ Rational exponents  $(-125)^{\frac{2}{3}}$