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}}