Cube Roots and Nth Roots
Learning Outcomes
- Define and simplify cube roots
- Define and evaluate nth roots
- Estimate roots that are not perfect
Example
Evaluate the following:- [latex] \sqrt[3]{-8}[/latex]
- [latex] \sqrt[3]{27}[/latex]
- [latex]0[/latex]
Answer: 1. We want to find a number whose cube is [latex]-8[/latex]. We know [latex]2[/latex] is the cube root of [latex]8[/latex], so maybe we can try [latex]-2[/latex] which gives [latex]-2\cdot{-2}\cdot{-2}=-8[/latex], so the cube root of [latex]-8[/latex] is [latex]-2[/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number. 2. We want to find a number whose cube is [latex]27[/latex]. [latex]3\cdot{3}\cdot{3}=27[/latex], so the cube root of [latex]27[/latex] is [latex]3[/latex]. 3. We want to find a number whose cube is [latex]0[/latex]. [latex]0\cdot0\cdot0[/latex], no matter how many times you multiply [latex]0[/latex] by itself, you will always get [latex]0[/latex].
Example
Simplify. [latex] \sqrt[3]{125}[/latex]Answer: [latex]125[/latex] ends in [latex]5[/latex], so you know that [latex]5[/latex] is a factor. Expand [latex]125[/latex] into [latex]5\cdot25[/latex].
[latex] \sqrt[3]{5\cdot 25}[/latex]
Factor [latex]25[/latex] into [latex]5[/latex]and [latex]5[/latex].[latex] \sqrt[3]{5\cdot 5\cdot 5}[/latex]
The factors are [latex]5\cdot5\cdot5[/latex], or [latex]5^{3}[/latex].[latex] \sqrt[3]{{{5}^{3}}}[/latex]
Answer
[latex-display] \sqrt[3]{125}=5[/latex-display]Example
Simplify. [latex] \sqrt[3]{32{{m}^{5}}}[/latex]Answer: Factor [latex]32[/latex] into prime factors.
[latex] \sqrt[3]{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot {{m}^{5}}}[/latex]
Since you are looking for the cube root, you need to find factors that appear [latex]3[/latex] times under the radical. Rewrite [latex] 2\cdot 2\cdot 2[/latex] as [latex] {{2}^{3}}[/latex].[latex] \sqrt[3]{{{2}^{3}}\cdot 2\cdot 2\cdot {{m}^{5}}}[/latex]
Rewrite [latex] {{m}^{5}}[/latex] as [latex] {{m}^{3}}\cdot {{m}^{2}}[/latex].[latex] \sqrt[3]{{{2}^{3}}\cdot 2\cdot 2\cdot {{m}^{3}}\cdot {{m}^{2}}}[/latex]
Rewrite the expression as a product of multiple radicals.[latex] \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{2\cdot 2}\cdot \sqrt[3]{{{m}^{3}}}\cdot \sqrt[3]{{{m}^{2}}}[/latex]
Simplify and multiply.[latex] 2\cdot \sqrt[3]{4}\cdot m\cdot \sqrt[3]{{{m}^{2}}}[/latex]
Answer
[latex-display] \sqrt[3]{32{{m}^{5}}}=2m\sqrt[3]{4{{m}^{2}}}[/latex-display]Example
Simplify. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]Answer: Factor the expression into cubes. Separate the cubed factors into individual radicals.
[latex]\begin{array}{r}\sqrt[3]{-1\cdot 27\cdot {{x}^{4}}\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}\cdot {{(3)}^{3}}\cdot {{x}^{3}}\cdot x\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{(3)}^{3}}}\cdot \sqrt[3]{{{x}^{3}}}\cdot \sqrt[3]{x}\cdot \sqrt[3]{{{y}^{3}}}\end{array}[/latex]
Simplify the cube roots.[latex] -1\cdot 3\cdot x\cdot y\cdot \sqrt[3]{x}[/latex]
Answer
[latex-display] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\sqrt[3]{x}[/latex-display][latex] \begin{array}{l}\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\\-3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\\-27\cdot {{x}^{3}}\cdot {{y}^{3}}\cdot \sqrt[3]{{{x}^{3}}}\\-27{{x}^{3}}{{y}^{3}}\cdot x\\-27{{x}^{4}}{{y}^{3}}\end{array}[/latex]
You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals [latex] \sqrt[3]{-81},\ \sqrt[5]{-64}[/latex], and [latex] \sqrt[7]{-2187}[/latex], but you cannot simplify the radicals [latex] \sqrt[{}]{-100},\ \sqrt[4]{-16}[/latex], or [latex] \sqrt[6]{-2,500}[/latex]. Let’s look at another example.Example
Simplify. [latex] \sqrt[3]{-24{{a}^{5}}}[/latex]Answer: Factor [latex]−24[/latex] to find perfect cubes. Here, [latex]−1[/latex] and 8 are the perfect cubes.
[latex] \sqrt[3]{-1\cdot 8\cdot 3\cdot {{a}^{5}}}[/latex]
Factor variables. You are looking for cube exponents, so you factor [latex]a^{5}[/latex] into [latex]a^{3}[/latex] and [latex]a^{2}[/latex].[latex] \sqrt[3]{{{(-1)}^{3}}\cdot {{2}^{3}}\cdot 3\cdot {{a}^{3}}\cdot {{a}^{2}}}[/latex]
Separate the factors into individual radicals.[latex] \sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{{{a}^{3}}}\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]
Simplify, using the property [latex] \sqrt[3]{{{x}^{3}}}=x[/latex].[latex] -1\cdot 2\cdot a\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]
This is the simplest form of this expression; all cubes have been pulled out of the radical expression.[latex] -2a\sqrt[3]{3{{a}^{2}}}[/latex]
Answer
[latex-display] \sqrt[3]{-24{{a}^{5}}}=-2a\sqrt[3]{3{{a}^{2}}}[/latex-display]Try It
[ohm_question]196049[/ohm_question]Example
Approximate [latex] \sqrt[3]{30}[/latex] and also find its value using a calculator.Answer: Find the cubes that surround [latex]30[/latex]. [latex]30[/latex] is in between the perfect cubes [latex]27[/latex] and [latex]81[/latex]. [latex] \sqrt[3]{27}=3[/latex] and [latex] \sqrt[3]{81}=4[/latex], so [latex] \sqrt[3]{30}[/latex] is between [latex]3[/latex] and [latex]4[/latex]. Use a calculator.
[latex]\sqrt[3]{30}\approx3.10723[/latex]
Try It
[ohm_question]189478[/ohm_question]Nth Roots
We learned above that the cube root of a number is written with a small number [latex]3[/latex], which looks like [latex] \sqrt[3]{a}[/latex]. This number placed just outside and above the radical symbol and is called the index. This little [latex]3[/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol. We can apply the same idea to any exponent and its corresponding root. 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 [latex]2[/latex]. In the radical expression, [latex]n[/latex] is called the index of the radical.Definition: 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
Evaluate each of the following:- [latex]\sqrt[5]{-32}[/latex]
- [latex]\sqrt[4]{81}[/latex]
- [latex]\sqrt[8]{-1}[/latex]
Answer:
- [latex]\sqrt[5]{-32}[/latex] Factor [latex]32[/latex], which gives [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
- [latex]\sqrt[4]{81}[/latex]. Factoring can help. We know that [latex]9\cdot9=81[/latex] and we can further factor each [latex]9[/latex]: [latex]\sqrt[4]{81}=\sqrt[4]{3\cdot3\cdot3\cdot3}=\sqrt[4]{3^4}=3[/latex]
- [latex]\sqrt[8]{-1}[/latex]. Since we have an [latex]8[/latex]th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1=+1[/latex]
Simplifying a radical
When working with exponents and radicals:- If n is odd, [latex] \sqrt[n]{{{x}^{n}}}=x[/latex].
- If n is even, [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex]. (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)
Try It
[ohm_question]94221[/ohm_question]Summary
A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd, and [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions. Nth roots can be approximated using trial and error or a calculator.Contribute!
Licenses & Attributions
CC licensed content, Original
- Screenshot: Rubik's Cube. Provided by: Lumen Learning License: CC BY: Attribution.
- Svreenshot: Caution. Provided by: Lumen Learning License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Simplify Cube Roots (Perfect Cube Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Cube Roots (Not Perfect Cube Radicands). Authored by: James Sousa (Mathispower4u.com) for 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 and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.