Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[/latex] is even, then [latex]a[/latex] cannot be negative.

[latex]{a}^{\frac{1}{n}}=\sqrt[n]{a}[/latex]

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

[latex]{a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}[/latex]

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Example 11: Rational Exponents

Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

[latex]{a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}[/latex]

How To: Given an expression with a rational exponent, write the expression as a radical.

  1. Determine the power by looking at the numerator of the exponent.
  2. Determine the root by looking at the denominator of the exponent.
  3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

Example 11: Writing Rational Exponents as Radicals

Write [latex]{343}^{\frac{2}{3}}[/latex] as a radical. Simplify.

Solution

The 2 tells us the power and the 3 tells us the root.

[latex]{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}=\sqrt[3]{{343}^{2}}[/latex]

We know that [latex]\sqrt[3]{343}=7[/latex] because [latex]{7}^{3}=343[/latex]. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

[latex]{343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}={7}^{2}=49[/latex]

Try It 11

Write [latex]{9}^{\frac{5}{2}}[/latex] as a radical. Simplify.

Solution

Example 12: Writing Radicals as Rational Exponents

Write [latex]\frac{4}{\sqrt[7]{{a}^{2}}}[/latex] using a rational exponent.

Solution

The power is 2 and the root is 7, so the rational exponent will be [latex]\frac{2}{7}[/latex]. We get [latex]\frac{4}{{a}^{\frac{2}{7}}}[/latex]. Using properties of exponents, we get [latex]\frac{4}{\sqrt[7]{{a}^{2}}}=4{a}^{\frac{-2}{7}}[/latex].

Try It 12

Write [latex]x\sqrt{{\left(5y\right)}^{9}}[/latex] using a rational exponent.

Solution

Example 13: Simplifying Rational Exponents

Simplify:

  1. [latex]5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)[/latex]
  2. [latex]{\left(\frac{16}{9}\right)}^{-\frac{1}{2}}[/latex]

Solution

  1. [latex]\begin{array}{cc}30{x}^{\frac{3}{4}}{x}^{\frac{1}{5}}\hfill & \text{Multiply the coefficients}.\\hfill \\ 30{x}^{\frac{3}{4}+\frac{1}{5}}\hfill & \text{Use properties of exponents}.\hfill \\ 30{x}^{\frac{19}{20}}\hfill & \text{Simplify}.\hfill \end{array}[/latex]
  2.  [latex]\begin{array}{cc}{\left(\frac{9}{16}\right)}^{\frac{1}{2}}\hfill & \text{ }\text{Use definition of negative exponents}.\hfill \\ \sqrt{\frac{9}{16}}\hfill & \text{ }\text{Rewrite as a radical}.\hfill \\ \frac{\sqrt{9}}{\sqrt{16}}\hfill & \text{ }\text{Use the quotient rule}.\hfill \\ \frac{3}{4}\hfill & \text{ }\text{Simplify}.\hfill \end{array}[/latex]

Try It 13

Simplify [latex]{\left(8x\right)}^{\frac{1}{3}}\left(14{x}^{\frac{6}{5}}\right)[/latex].

Solution