Non-Linear Equations: Background You’ll Need 1

  • Simplify and rewrite rational exponents.

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]x[/latex] cannot be negative.

Radical Form Exponent Form
[latex]\sqrt{x}[/latex] [latex]x^{\frac{1}{2}}[/latex]
[latex]\sqrt[3]{x}[/latex] [latex]x^{\frac{1}{3}}[/latex]
[latex]\sqrt[4]{x}[/latex] [latex]x^{\frac{1}{4}}[/latex]
… …
[latex]\sqrt[n]{x}[/latex] [latex]x^{\frac{1}{n}}[/latex]
We can also have rational exponents with numerators other than 1.

rational exponents

Rational exponents are another way to express principal [latex]\text{n}^{\text{th}}[/latex] roots.

 

The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

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

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

When a base has a negative exponent, you can rewrite it as the reciprocal of the base with a positive exponent. In mathematical terms:

[latex]a^{-x} = \frac{1}{a^x}[/latex]

This rule helps simplify expressions where negative exponents are present by turning them into fractions with positive exponents.

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

Simplify:

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