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):

Example: 

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

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.

Examples:

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