We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

TEXT
Unlock Solution Steps
Sign in to
Symbolab
Get full access to all Solution Steps for any math problem
By continuing, you agree to our Terms of Use
and have read our Privacy Policy

Study Guides > ALGEBRA / TRIG I

Cube Roots and Nth Roots

Learning Outcomes

  • Define and simplify cube roots
  • Define and evaluate nth roots
  • Estimate roots that are not perfect
Rubik's cube Rubik's Cune
We know that [latex]5^2=25, \text{ and }\sqrt{25}=5[/latex], but what if we want to "undo" [latex]5^3=125, \text{ or }5^4=625[/latex]? We can use higher order roots to answer these questions. While square roots are probably the most common radical, you can also find the third root, the fifth root, the [latex]10th[/latex] root, or really any other nth root of a number. Just as the square root is a number that, when squared, gives the radicand, the cube root is a number that, when cubed, gives the radicand. The cube root of a number is written with a small number  [latex]3[/latex], called the index, just outside and above the radical symbol. It looks like [latex] \sqrt[3]{{}}[/latex]. This little [latex]3[/latex] distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.
CautionCaution! Be careful to distinguish between [latex] \sqrt[3]{x}[/latex], the cube root of x, and [latex] 3\sqrt{x}[/latex], three times the square root of x. They may look similar at first, but they lead you to much different expressions!
Suppose we know that [latex]{a}^{3}=8[/latex]. We want to find what number raised to the [latex]3[/latex]rd power is equal to [latex]8[/latex]. Since [latex]{2}^{3}=8[/latex], we say that [latex]2[/latex] is the cube root of [latex]8[/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.

Example

Evaluate the following:
  1. [latex] \sqrt[3]{-8}[/latex]
  2. [latex] \sqrt[3]{27}[/latex]
  3. [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].

As we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]−8[/latex]? [latex] \sqrt[3]{-8}=-2[/latex] because [latex] -2\cdot -2\cdot -2=-8[/latex]. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider [latex] \sqrt[3]{{{(-1)}^{3}}}=-1[/latex]. We can also use factoring to simplify cube roots such as [latex] \sqrt[3]{125}[/latex]. You can read this as “the third root of [latex]125[/latex]” or “the cube root of [latex]125[/latex].” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals [latex]125[/latex]. Let’s factor [latex]125[/latex] and find that number.

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]

The prime factors of [latex]125[/latex] are [latex]5\cdot5\cdot5[/latex], which can be rewritten as [latex]5^{3}[/latex]. The cube root of a cubed number is the number itself, so [latex] \sqrt[3]{{{5}^{3}}}=5[/latex]. You have found the cube root, the three identical factors that when multiplied together give [latex]125[/latex]. [latex]125[/latex] is known as a perfect cube because its cube root is an integer. In the following video, we show more examples of finding a cube root. https://youtu.be/9Nh-Ggd2VJo Here’s an example of how to simplify a radical that is not a perfect cube.

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]

In the example below, we use the idea that  [latex] \sqrt[3]{{{(-1)}^{3}}}=-1[/latex] to simplify the radical.  You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.

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]

You could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\sqrt[3]{x}[/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[/latex].

[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]

In the next video, we share examples of finding cube roots with negative radicands.

Try It

[ohm_question]196049[/ohm_question]
https://youtu.be/BtJruOpmHCE In the same way that we learned earlier that we can estimate square roots, we can also estimate cube roots.

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:
  1. [latex]\sqrt[5]{-32}[/latex]
  2. [latex]\sqrt[4]{81}[/latex]
  3. [latex]\sqrt[8]{-1}[/latex]

Answer:

  1. [latex]\sqrt[5]{-32}[/latex] Factor [latex]32[/latex], which gives [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
  2. [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]
  3. [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]

The steps to consider when simplifying a radical are outlined below.

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.)
In the following video, we show more examples of how to evaluate nth roots. https://youtu.be/vA2DkcUSRSk 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 evaluate the radicals [latex] \sqrt[3]{-81},\ \sqrt[5]{-64}[/latex], and [latex] \sqrt[7]{-2187}[/latex] because the all have an odd numbered index, but you cannot evaluate the radicals [latex] \sqrt[{}]{-100},\ \sqrt[4]{-16}[/latex], or [latex] \sqrt[6]{-2,500}[/latex] because they all have an even numbered index.

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!

Did you have an idea for improving this content? We’d love your input.

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.