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Fractional Exponents

Sometimes exponents are written as fractions.  Fractional exponents contain two pieces of information:

  • The exponent that the base should be raised to (the numerator of the fraction).
  • The index of the root that that should be applied to the base and exponent (the denominator of the fraction).

For example, here are a set of fractional exponents written as radicals:

$\eqalign{4^{\frac{2}{3}}&=\sqrt[3]{4^2}\\2^{\frac{3}{2}}&=\sqrt{2^3}\quad\text{Note: most square roots have an index of 2, so we don't write it}\\5^{\frac{6}{5}}&=\sqrt[5]{5^6}}$

You can simplify a radical that comes from a fractional equation just like you can simplify any other radical (see lesson on Simplifying Radicals to review):


$\eqalign{4^{\frac{2}{3}}&=\sqrt[3]{4^2}\\&= \sqrt[3]{4\cdot 4}\\&=\sqrt[3]{2\cdot2 \cdot 2 \cdot 2}\\&=\sqrt[3]{\boxed {2 \cdot 2 \cdot 2} \cdot 2}\\&= 2\sqrt[3]{2}}$

Fractional exponents also follow all of the same rules as other exponents.  To review those rules:

  • When multipying exponents with the same base: add the exponents.
  • When finding an exponent of an exponent: multiply the exponents.
  • When dividing exponents with the same base: subtract the exponents.


$\eqalign{\text{Multiplying exponents with the same base:}\quad(x^{\frac{2}{3}})(x^{\frac{4}{5}})&=x^{\frac{22}{15}}=\sqrt[15]{x^{22}}=x\sqrt[15]{x^7}\\\text{Exponents of exponents:}\quad(x^{\frac{2}{3}})^{\frac{4}{3}}&=x^{\frac{8}{9}}\\\text{Dividing exponents with the same base:}\quad\frac{x^{\frac{1}{2}}}{x^{\frac{1}{4}}}&=x^2}$

Practice Problems:

  • Fractional Exponents

    Write the fractional exponents in radical form.

    1. $4^{\frac{2}{3}}$
    2. $a^{\frac{4}{5}}$
    3. $x^{\frac{1}{2}}y^{\frac{3}{4}}$

    Simplify fractional exponents.

    1. $(4^{\frac{2}{3}})(4^{\frac{1}{3}})$
    2. $(2^{\frac{4}{5}})^{\frac{1}{2}}$
    3. $1^{\frac{1}{2}}3^{\frac{3}{4}}$
    4. $(x^{\frac{2}{5}})^{\frac{2}{3}}$
    5. $(3^2)^{\frac{2}{3}}$
    6. $\dfrac{x^{\frac{1}{4}}}{x^{\frac{1}{2}}}$
    7. $\dfrac{x^{\frac{1}{8}}y^{\frac{2}{3}}}{x^{\frac{1}{4}}y^{\frac{1}{3}}}$
    8. $\dfrac{2^{\frac{5}{6}}}{2^{\frac{2}{3}}}$
    9. $\dfrac{3^{\frac{2}{3}}}{3^{\frac{4}{3}}}$

    Answer Key: