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