Fractional Indices

Fractional indices can be expressed as roots. Here are some basic rules:

  • a^{\frac{1}{2}}=\sqrt{a} (square root)
  • a^{\frac{1}{3}}=\sqrt[3]{a} (cube root)
  • a^{\frac{1}{n}}=\sqrt[n]{a} (nth root)
  • a^{\frac{m}{n}}=\left(a^{m}\right)^{\frac{1}{n}}=\sqrt[n]{a^{m}}

Examples

Simplify the following expressions:

1. 8^{\frac{2}{3}}

2. 16^{\frac{1}{4}}

3. 125^{-\frac{2}{3}}

4. 64^{-\frac{4}{3}}

5. 36^{-\frac{1}{2}}

6. 81^{\frac{5}{4}}

7. \left(\frac{2}{3}\right)^{-2}

8. -8^{\frac{2}{3}}

9. -27^{-\frac{4}{3}}

1. 8^{\frac{2}{3}}=\left(2^{3}\right)^{\frac{2}{3}}=2^{2}=48^{\frac{2}{3}}=\left(2^{3}\right)^{\frac{2}{3}}=2^{2}=4

2. 16^{\frac{1}{4}}=\left(2^{4}\right)^{\frac{1}{4}}=2^{1}=216^{\frac{1}{4}}=\left(2^{4}\right)^{\frac{1}{4}}=2^{1}=2

3. 125^{-\frac{2}{3}}=\left(5^{3}\right)^{-\frac{2}{3}}=5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}125^{-\frac{2}{3}}=\left(5^{3}\right)^{-\frac{2}{3}}=5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}

4. 64^{-\frac{4}{3}}=\left(2^{6}\right)^{-\frac{4}{3}}=2^{-4}=\frac{1}{2^{4}}=\frac{1}{16}64^{-\frac{4}{3}}=\left(2^{6}\right)^{-\frac{4}{3}}=2^{-4}=\frac{1}{2^{4}}=\frac{1}{16}

5. 36^{-\frac{1}{2}}=\left(6^{2}\right)^{-\frac{1}{2}}=6^{-1}=\frac{1}{6}36^{-\frac{1}{2}}=\left(6^{2}\right)^{-\frac{1}{2}}=6^{-1}=\frac{1}{6}

6. 81^{\frac{5}{4}}=\left(3^{4}\right)^{\frac{5}{4}}=3^{5}=24381^{\frac{5}{4}}=\left(3^{4}\right)^{\frac{5}{4}}=3^{5}=243

7. \left(\frac{2}{3}\right)^{-2}=\frac{1}{\left(\frac{2}{3}\right)^{2}}=\frac{1}{\frac{4}{9}}=1\div \frac{4}{9}=1\times \frac{9}{4}=\frac{9}{4}\left(\frac{2}{3}\right)^{-2}=\frac{1}{\left(\frac{2}{3}\right)^{2}}=\frac{1}{\frac{4}{9}}=1\div \frac{4}{9}=1\times \frac{9}{4}=\frac{9}{4}

Note that, \left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^{m}\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^{m}

8. -8^{\frac{2}{3}}=\left[\left(-2\right)^{3}\right]^{\frac{2}{3}}=\left(-2\right)^{2}=4-8^{\frac{2}{3}}=\left[\left(-2\right)^{3}\right]^{\frac{2}{3}}=\left(-2\right)^{2}=4

9. -27^{-\frac{4}{3}}=\left[\left(-3\right)^{3}\right]^{-\frac{4}{3}}=\left(-3\right)^{-4}=\frac{1}{\left(-3\right)^{4}}=\frac{1}{81}-27^{-\frac{4}{3}}=\left[\left(-3\right)^{3}\right]^{-\frac{4}{3}}=\left(-3\right)^{-4}=\frac{1}{\left(-3\right)^{4}}=\frac{1}{81}

Example

Evaluate the following:

1. 16^{-\frac{3}{4}}

2. 125^{-\frac{1}{3}}

3. \left(\frac{9}{16}\right)^{-\frac{3}{2}}

4. \left(2\frac{1}{4}\right)^3

5. \left(2\frac{1}{4}\right)^{-3}

6. \left(2\frac{1}{4}\right)^{\frac{3}{2}}

7. \left(2\frac{1}{4}\right)^{-\frac{3}{2}}

1. 16^{-\frac{3}{4}}=\left(2^{4}\right)^{-\frac{3}{4}}=2^{-3}=\frac{1}{2^3}=\frac{1}{8}16^{-\frac{3}{4}}=\left(2^{4}\right)^{-\frac{3}{4}}=2^{-3}=\frac{1}{2^3}=\frac{1}{8}

2. 125^{-\frac{1}{3}}=\left(5^{3}\right)^{-\frac{1}{3}}=5^{-1}=\frac{1}{5}125^{-\frac{1}{3}}=\left(5^{3}\right)^{-\frac{1}{3}}=5^{-1}=\frac{1}{5}

3. \left(\frac{9}{16}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{4}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{4}\right)^{-3}=\frac{1}{\left(\frac{3}{4}\right)^3}=\frac{1}{\frac{27}{64}}=\frac{64}{27}\left(\frac{9}{16}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{4}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{4}\right)^{-3}=\frac{1}{\left(\frac{3}{4}\right)^3}=\frac{1}{\frac{27}{64}}=\frac{64}{27}

4. \left(2\frac{1}{4}\right)^3=\left(\frac{9}{4}\right)^3=\frac{729}{64}\left(2\frac{1}{4}\right)^3=\left(\frac{9}{4}\right)^3=\frac{729}{64}

5. \left(2\frac{1}{4}\right)^{-3}=\left(\frac{9}{4}\right)^{-3}=\left(\frac{4}{9}\right)^3=\frac{64}{729}\left(2\frac{1}{4}\right)^{-3}=\left(\frac{9}{4}\right)^{-3}=\left(\frac{4}{9}\right)^3=\frac{64}{729}

6. \left(2\frac{1}{4}\right)^{\frac{3}{2}}=\left(\frac{9}{4}\right)^{\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{\frac{3}{2}}=\left(\frac{3}{2}\right)^3=\frac{27}{8}\left(2\frac{1}{4}\right)^{\frac{3}{2}}=\left(\frac{9}{4}\right)^{\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{\frac{3}{2}}=\left(\frac{3}{2}\right)^3=\frac{27}{8}

7. \left(2\frac{1}{4}\right)^{-\frac{3}{2}}=\left(\frac{9}{4}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{2}\right)^{-3}=\left(\frac{2}{3}\right)^3=\frac{8}{27}\left(2\frac{1}{4}\right)^{-\frac{3}{2}}=\left(\frac{9}{4}\right)^{-\frac{3}{2}}=\left(\left(\frac{3}{2}\right)^2\right)^{-\frac{3}{2}}=\left(\frac{3}{2}\right)^{-3}=\left(\frac{2}{3}\right)^3=\frac{8}{27}

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