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

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 52=25, and 25=55^2=25, \text{ and }\sqrt{25}=5, but what if we want to "undo" 53=125, or 54=6255^3=125, \text{ or }5^4=625? 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 10th10th 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  33, called the index, just outside and above the radical symbol. It looks like 3 \sqrt[3]{{}}. This little 33 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 x3 \sqrt[3]{x}, the cube root of x, and 3x 3\sqrt{x}, 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 a3=8{a}^{3}=8. We want to find what number raised to the 33rd power is equal to 88. Since 23=8{2}^{3}=8, we say that 22 is the cube root of 88. In the next example, we will evaluate the cube roots of some perfect cubes.

Example

Evaluate the following:
  1. 83 \sqrt[3]{-8}
  2. 273 \sqrt[3]{27}
  3. 00

Answer: 1. We want to find a number whose cube is 8-8. We know 22 is the cube root of 88, so maybe we can try 2-2 which gives 222=8-2\cdot{-2}\cdot{-2}=-8, so the cube root of 8-8 is 2-2. 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 2727. 333=273\cdot{3}\cdot{3}=27, so the cube root of 2727 is 33. 3.  We want to find a number whose cube is 00. 0000\cdot0\cdot0, no matter how many times you multiply 00 by itself, you will always get 00.

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 8−8? 83=2 \sqrt[3]{-8}=-2 because 222=8 -2\cdot -2\cdot -2=-8. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider (1)33=1 \sqrt[3]{{{(-1)}^{3}}}=-1. We can also use factoring to simplify cube roots such as 1253 \sqrt[3]{125}. You can read this as “the third root of 125125” or “the cube root of 125125.” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125125. Let’s factor 125125 and find that number.

Example

Simplify. 1253 \sqrt[3]{125}

Answer: 125125 ends in 55, so you know that  55 is a factor. Expand 125125 into 5255\cdot25.

5253 \sqrt[3]{5\cdot 25}

Factor 2525 into 55and 55.

5553 \sqrt[3]{5\cdot 5\cdot 5}

The factors are 5555\cdot5\cdot5, or 535^{3}.

533 \sqrt[3]{{{5}^{3}}}

Answer

1253=5 \sqrt[3]{125}=5

The prime factors of 125125 are 5555\cdot5\cdot5, which can be rewritten as 535^{3}. The cube root of a cubed number is the number itself, so 533=5 \sqrt[3]{{{5}^{3}}}=5. You have found the cube root, the three identical factors that when multiplied together give 125125. 125125 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. 32m53 \sqrt[3]{32{{m}^{5}}}

Answer: Factor 3232 into prime factors.

22222m53 \sqrt[3]{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot {{m}^{5}}}

Since you are looking for the cube root, you need to find factors that appear 33 times under the radical. Rewrite 222 2\cdot 2\cdot 2 as 23 {{2}^{3}}.

2322m53 \sqrt[3]{{{2}^{3}}\cdot 2\cdot 2\cdot {{m}^{5}}}

Rewrite m5 {{m}^{5}} as m3m2 {{m}^{3}}\cdot {{m}^{2}}.

2322m3m23 \sqrt[3]{{{2}^{3}}\cdot 2\cdot 2\cdot {{m}^{3}}\cdot {{m}^{2}}}

Rewrite the expression as a product of multiple radicals.

233223m33m23 \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{2\cdot 2}\cdot \sqrt[3]{{{m}^{3}}}\cdot \sqrt[3]{{{m}^{2}}}

Simplify and multiply.

243mm23 2\cdot \sqrt[3]{4}\cdot m\cdot \sqrt[3]{{{m}^{2}}}

Answer

32m53=2m4m23 \sqrt[3]{32{{m}^{5}}}=2m\sqrt[3]{4{{m}^{2}}}

In the example below, we use the idea that  (1)33=1 \sqrt[3]{{{(-1)}^{3}}}=-1 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. 27x4y33 \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}

Answer: Factor the expression into cubes. Separate the cubed factors into individual radicals.

127x4y33(1)3(3)3x3xy33(1)33(3)33x33x3y33\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}

Simplify the cube roots.

13xyx3 -1\cdot 3\cdot x\cdot y\cdot \sqrt[3]{x}

Answer

27x4y33=3xyx3 \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\sqrt[3]{x}

You could check your answer by performing the inverse operation. If you are right, when you cube 3xyx3 -3xy\sqrt[3]{x} you should get 27x4y3 -27{{x}^{4}}{{y}^{3}}.

(3xyx3)(3xyx3)(3xyx3)333xxxyyyx3x3x327x3y3x3327x3y3x27x4y3 \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}

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 813, 645 \sqrt[3]{-81},\ \sqrt[5]{-64}, and 21877 \sqrt[7]{-2187}, but you cannot simplify the radicals 100, 164 \sqrt[{}]{-100},\ \sqrt[4]{-16}, or 2,5006 \sqrt[6]{-2,500}. Let’s look at another example.

Example

Simplify. 24a53 \sqrt[3]{-24{{a}^{5}}}

Answer: Factor 24−24 to find perfect cubes. Here, 1−1 and 8 are the perfect cubes.

183a53 \sqrt[3]{-1\cdot 8\cdot 3\cdot {{a}^{5}}}

Factor variables. You are looking for cube exponents, so you factor a5a^{5} into a3a^{3} and a2a^{2}.

(1)3233a3a23 \sqrt[3]{{{(-1)}^{3}}\cdot {{2}^{3}}\cdot 3\cdot {{a}^{3}}\cdot {{a}^{2}}}

Separate the factors into individual radicals.

(1)33233a333a23 \sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{{{a}^{3}}}\cdot \sqrt[3]{3\cdot {{a}^{2}}}

Simplify, using the property x33=x \sqrt[3]{{{x}^{3}}}=x. 

12a3a23 -1\cdot 2\cdot a\cdot \sqrt[3]{3\cdot {{a}^{2}}}

This is the simplest form of this expression; all cubes have been pulled out of the radical expression.

2a3a23 -2a\sqrt[3]{3{{a}^{2}}}

Answer

24a53=2a3a23 \sqrt[3]{-24{{a}^{5}}}=-2a\sqrt[3]{3{{a}^{2}}}

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 303 \sqrt[3]{30} and also find its value using a calculator.

Answer: Find the cubes that surround 3030. 3030 is in between the perfect cubes 2727 and 8181. 273=3 \sqrt[3]{27}=3 and 813=4 \sqrt[3]{81}=4, so 303 \sqrt[3]{30} is between 33 and 44. Use a calculator.

3033.10723\sqrt[3]{30}\approx3.10723

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 33, which looks like a3 \sqrt[3]{a}. This number placed just outside and above the radical symbol and is called the index. This little 33 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 aa is a number that, when raised to the nth power, gives aa. For example, 33 is the 5th root of 243243 because (3)5=243{\left(3\right)}^{5}=243. If aa is a real number with at least one nth root, then the principal nth root of aa is the number with the same sign as aa that, when raised to the nth power, equals aa. The principal nth root of aa is written as an\sqrt[n]{a}, where nn is a positive integer greater than or equal to 22. In the radical expression, nn is called the index of the radical.

Definition: Principal nth Root

If aa is a real number with at least one nth root, then the principal nth root of aa, written as an\sqrt[n]{a}, is the number with the same sign as aa that, when raised to the nth power, equals aa. The index of the radical is nn.

Example

Evaluate each of the following:
  1. 325\sqrt[5]{-32}
  2. 814\sqrt[4]{81}
  3. 18\sqrt[8]{-1}

Answer:

  1. 325\sqrt[5]{-32} Factor 3232, which gives (2)5=32 {\left(-2\right)}^{5}=-32 \\ \text{ }
  2. 814\sqrt[4]{81}. Factoring can help. We know that 99=819\cdot9=81 and we can further factor each 99: 814=33334=344=3\sqrt[4]{81}=\sqrt[4]{3\cdot3\cdot3\cdot3}=\sqrt[4]{3^4}=3
  3. 18\sqrt[8]{-1}. Since we have an 88th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, 11111111=+1-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1=+1

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

Simplifying a radical

When working with exponents and radicals:
  • If n is odd, xnn=x \sqrt[n]{{{x}^{n}}}=x.
  • If n is even, xnn=x \sqrt[n]{{{x}^{n}}}=\left| x \right|. (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 813, 645 \sqrt[3]{-81},\ \sqrt[5]{-64}, and 21877 \sqrt[7]{-2187} because the all have an odd numbered index, but you cannot evaluate the radicals 100, 164 \sqrt[{}]{-100},\ \sqrt[4]{-16}, or 2,5006 \sqrt[6]{-2,500} 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 xnn=x \sqrt[n]{{{x}^{n}}}=x if n is odd, and xnn=x \sqrt[n]{{{x}^{n}}}=\left| x \right| 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