Rational Indices and Roots |
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Rational Indices
\( a^{\frac{1}{n}} \)
Rule: \( a^{\frac{1}{n}} = \sqrt[n]{a} \).
Example: \( 16^{\frac{1}{2}} = \sqrt{16} = 4 \).
Example: \( 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 \).
\( a^{\frac{m}{n}} \)
Rule: \( a^{\frac{m}{n}} = \Bigl(a^m\Bigr)^{\frac{1}{n}} = \Bigl(a^{\frac{1}{n}}\Bigr)^m \).
Example: \( 9^{\frac{3}{2}} = \sqrt{9^3} = \sqrt{729} = 27 \).
Example: \( 16^{\frac{3}{2}} = \sqrt{16^3} = \sqrt{4096} = 64 \).
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