Radical Expressions and Rational Exponents
Learning Outcomes
- Convert radicals to expressions with rational exponents
- Convert expressions with rational exponents to their radical equivalent
Write an Expression with a Rational Exponent as a Radical
Radicals and fractional exponents are alternate ways of expressing the same thing. In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
[latex] \sqrt{16}[/latex] | [latex] {{16}^{\tfrac{1}{2}}}[/latex] | [latex]4[/latex] |
[latex] \sqrt{25}[/latex] | [latex] {{25}^{\tfrac{1}{2}}}[/latex] | [latex]5[/latex] |
[latex] \sqrt{100}[/latex] | [latex] {{100}^{\tfrac{1}{2}}}[/latex] | [latex]10[/latex] |
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
[latex] \sqrt[3]{8}[/latex] | [latex] {{8}^{\tfrac{1}{3}}}[/latex] | [latex]2[/latex] |
[latex] \sqrt[3]{125}[/latex] | [latex] {{125}^{\tfrac{1}{3}}}[/latex] | [latex]5[/latex] |
[latex] \sqrt[3]{1000}[/latex] | [latex] {{1000}^{\tfrac{1}{3}}}[/latex] | [latex]10[/latex] |
Radical Form |
Exponent Form |
---|---|
[latex] \sqrt{x}[/latex] | [latex] {{x}^{\tfrac{1}{2}}}[/latex] |
[latex] \sqrt[3]{x}[/latex] | [latex] {{x}^{\tfrac{1}{3}}}[/latex] |
[latex] \sqrt[4]{x}[/latex] | [latex] {{x}^{\tfrac{1}{4}}}[/latex] |
… | … |
[latex] \sqrt[n]{x}[/latex] | [latex] {{x}^{\tfrac{1}{n}}}[/latex] |
Example
Express [latex] {{(2x)}^{^{\frac{1}{3}}}}[/latex] in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
[latex]\sqrt[3]{2x} [/latex]
The parentheses in [latex] {{\left( 2x \right)}^{\frac{1}{3}}}[/latex] indicate that the exponent refers to everything within the parentheses.Example
Express [latex] 2{{x}^{^{\frac{1}{3}}}}[/latex] in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
[latex] 2\sqrt[3]{x}[/latex]
The exponent refers only to the part of the expression immediately to the left of the exponent, in this case x, but not the [latex]2[/latex].Write a Radical Expression as an Expression with a Rational Exponent
Example
Write [latex] \sqrt[4]{81}[/latex] as an expression with a rational exponent.Answer: The radical form [latex] \Large\sqrt[4]{{\,\,\,\,}}[/latex] can be rewritten as the exponent [latex] \frac{1}{4}[/latex]. Remove the radical and place the exponent next to the base.
Example
Express [latex] 4\sqrt[3]{xy}[/latex] with rational exponents.Answer: Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is [latex]3[/latex], so the rational exponent will be [latex] \frac{1}{3}[/latex].
[latex] 4{{(xy)}^{\frac{1}{3}}}[/latex]
Since [latex]4[/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.Rational Exponents Whose Numerator is Not Equal to One
All of the numerators for the fractional exponents in the examples above were [latex]1[/latex]. You can use fractional exponents that have numerators other than [latex]1[/latex] to express roots, as shown below.
Radical |
Exponent |
---|---|
[latex] \sqrt{9}[/latex] | [latex]9^{\frac{1}{2}}[/latex] |
[latex] \sqrt[3]{{{9}^{2}}}[/latex] | [latex]9^{\frac{2}{3}}[/latex] |
[latex]\sqrt[4]{9^{3}}[/latex] | [latex]9^{\frac{3}{4}}[/latex] |
[latex]\sqrt[5]{9^{2}}[/latex] | [latex]9^{\frac{2}{5}}[/latex] |
… | … |
[latex]\sqrt[n]{9^{x}}[/latex] | [latex]9^{\frac{x}{n}}[/latex] |
Writing Rational Exponents
Any radical in the form [latex]\sqrt[n]{a^{x}}[/latex] can be written using a fractional exponent in the form [latex]a^{\frac{x}{n}}[/latex].Example
Rewrite the radicals using a rational exponent, then simplify your result.- [latex]\sqrt[3]{{{a}^{6}}}[/latex]
- [latex]\sqrt[12]{16^3}[/latex]
Answer: 1.[latex]\sqrt[n]{a^{x}}[/latex] can be rewritten as [latex]a^{\frac{x}{n}}[/latex], so in this case [latex]n=3,\text{ and }x=6[/latex], therefore [latex-display]\sqrt[3]{{{a}^{6}}}={{a}^{\frac{6}{3}}}[/latex-display] Simplify the exponent. [latex-display]{{a}^{\frac{6}{3}}}={{a}^{2}}[/latex-display] 2. [latex]\sqrt[n]{a^{x}}[/latex] can be rewritten as [latex]a^{\frac{x}{n}}[/latex], so in this case [latex]n=12,\text{ and }x=3[/latex], therefore
[latex]\sqrt[12]{16^3}={16}^{\frac{3}{12}}={16}^{\frac{1}{4}}[/latex]
Simplify the expression using rules for exponents. [latex-display]{16}^{\frac{1}{4}}=2[/latex-display]Example
Rewrite the expressions using a radical.- [latex]{x}^{\frac{2}{3}}[/latex]
- [latex]{5}^{\frac{4}{7}}[/latex]
Answer:
- [latex]{x}^{\frac{2}{3}}[/latex], the numerator is [latex]2[/latex] and the denominator is [latex]3[/latex], therefore we will have the third root of x squared, [latex]\sqrt[3]{x^2}[/latex]
- [latex]{5}^{\frac{4}{7}}[/latex], the numerator is [latex]4[/latex] and the denominator is [latex]7[/latex], so we will have the seventh root of [latex]5[/latex] raised to the fourth power. [latex]\sqrt[7]{5^4}[/latex]
Summary
Any radical in the form [latex]\sqrt[n]{a^{x}}[/latex] can be written using a fractional exponent in the form [latex]a^{\frac{x}{n}}[/latex]. Rewriting radicals using fractional exponents can be useful when simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.Licenses & Attributions
CC licensed content, Original
- Write Expressions Using Radicals and Rational Exponents. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Precalculus. Provided by: OpenStax Authored by: Abramson, Jay. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: Public Domain: No Known Copyright. License terms: Dwonload fro free at : http://cnx.org/contents/[email protected]:1/Preface.