3.3.1 – Using Rational Roots
Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.
Understanding nth Roots
Suppose we know that [latex]\,{a}^{3}=8.\,[/latex] We want to find what number raised to the 3rd power is equal to 8. Since [latex]\,{2}^{3}=8,[/latex] we say that 2 is the cube root of 8.
The nth root of [latex]\,a\,[/latex] is a number that, when raised to the nth power, gives [latex]\,a.\,[/latex] For example, [latex]\,-3\,[/latex] is the 5th root of [latex]\,-243\,[/latex] because [latex]\,{\left(-3\right)}^{5}=-243.\,[/latex] If [latex]\,a\,[/latex] is a real number with at least one nth root, then the principal nth root of [latex]\,a\,[/latex] is the number with the same sign as [latex]\,a\,[/latex] that, when raised to the nth power, equals [latex]\,a.[/latex]
The principal nth root of [latex]\,a\,[/latex] is written as [latex]\,\sqrt[n]{a},[/latex] where [latex]\,n\,[/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]\,n\,[/latex] is called the index of the radical.
Principal nth Root
If [latex]\,a\,[/latex] is a real number with at least one nth root, then the principal nth root of [latex]\,a,[/latex] written as [latex]\,\sqrt[n]{a},[/latex] is the number with the same sign as [latex]\,a\,[/latex] that, when raised to the nth power, equals [latex]\,a.\,[/latex] The index of the radical is [latex]\,n.[/latex]
Example 1 – Simplifying nth Roots
Simplify each of the following:
- [latex]\sqrt[5]{-32}[/latex]
- [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
- [latex]-\sqrt[3]{\frac{8{x}^{6}}{125}}[/latex]
- [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]
- [latex]\sqrt[5]{-32}=-2\,[/latex] because [latex]\,{\left(-2\right)}^{5}=-32[/latex]
- First, express the product as a single radical expression. [latex]\,\sqrt[4]{4,096}=8\,[/latex] because [latex]\,{8}^{4}=4,096[/latex]
- [latex]\begin{array}{cc}\frac{-\sqrt[3]{8{x}^{6}}}{\sqrt[3]{125}}\hfill & \phantom{\rule{3em}{0ex}}\text{Write as quotient of two radical expressions}.\hfill \\ \frac{-2{x}^{2}}{5}\hfill & \phantom{\rule{3em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
- [latex]\begin{array}{cc}8\sqrt[4]{3}-2\sqrt[4]{3}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify to get equal radicands}.\hfill \\ 6\sqrt[4]{3} \hfill & \phantom{\rule{2em}{0ex}}\text{Add}.\hfill \end{array}[/latex]
Try It
Simplify.
- [latex]\sqrt[3]{-216}[/latex]
- [latex]\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
- [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]
Show answer
- [latex]-6[/latex]
- [latex]6[/latex]
- [latex]88\sqrt[3]{9}[/latex]
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.
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.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
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
How To
Given an expression with a rational exponent, write the expression as a radical.
- Determine the power by looking at the numerator of the exponent.
- Determine the root by looking at the denominator of the exponent.
- Using the base as the radicand, raise the radicand to the power and use the root as the index.
Example 2 – Writing Rational Exponents as Radicals
Write [latex]\,{343}^{\frac{2}{3}}\,[/latex] as a radical. Simplify.
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
Write [latex]\,{9}^{\frac{5}{2}}\,[/latex] as a radical. Simplify.
Show answer
[latex]{\left(\sqrt{9}\right)}^{5}={3}^{5}=243[/latex]
Example 3 – Writing Radicals as Rational Exponents
Write [latex]\,\frac{4}{\sqrt[7]{{a}^{2}}}\,[/latex] using a rational exponent.
Try It
Write [latex]\,x\sqrt{{\left(5y\right)}^{9}}\,[/latex] using a rational exponent.
Show answer
[latex]x{\left(5y\right)}^{\frac{9}{2}}[/latex]
Example 4 – Simplifying Rational Exponents
Simplify:
- [latex]5\left(2{x}^{\frac{3}{4}}\right)\left(3{x}^{\frac{1}{5}}\right)[/latex]
- [latex]{\left(\frac{16}{9}\right)}^{-\frac{1}{2}}[/latex]
- [latex]\begin{array}{cc}30{x}^{\frac{3}{4}}{x}^{\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Multiply the coefficients}.\hfill \\ 30{x}^{\frac{3}{4}+\frac{1}{5}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Use properties of exponents}.\hfill \\ 30{x}^{\frac{19}{20}}\hfill & \phantom{\rule{2.5em}{0ex}}\text{Simplify}.\hfill \end{array}[/latex]
- [latex]\begin{array}{cc}{\left(\frac{9}{16}\right)}^{\frac{1}{2}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Use definition of negative exponents}.\hfill \\ \sqrt{\frac{9}{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Rewrite as a radical}.\hfill \\ \frac{\sqrt{9}}{\sqrt{16}}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Use the quotient rule}.\hfill \\ \frac{3}{4}\hfill & \phantom{\rule{2em}{0ex}}\text{ }\text{Simplify}.\hfill \end{array}[/latex]
Try It
Simplify [latex]\,\left[{\left(8x\right)}^{\frac{1}{3}}\right]\left(14{x}^{\frac{6}{5}}\right).[/latex]
Show answer
[latex]28{x}^{\frac{23}{15}}[/latex]
Access these online resources for additional instruction and practice with radicals and rational exponents.
Key Concepts
- The principal nth root of [latex]\,a\,[/latex] is the number with the same sign as [latex]\,a\,[/latex] that when raised to the nth power equals [latex]\,a.\,[/latex] These roots have the same properties as square roots. See Example 1.
- Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 2 and Example 3.
- The properties of exponents apply to rational exponents. See Example 4.
Glossary
- principal square root
- the nonnegative square root of a number [latex]\,a\,[/latex] that, when multiplied by itself, equals [latex]\,a[/latex]
- principal nth root
- the number with the same sign as [latex]\,a\,[/latex] that when raised to the nth power equals [latex]\,a[/latex]