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=5, but what if we want to “undo” 53=125, or 54=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 10th 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  3, called the index, just outside and above the radical symbol. It looks like 3. This little 3 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, the cube root of x, and 3x, 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. We want to find what number raised to the 3rd power is equal to 8. Since 23=8, we say that 2 is the cube root of 8. In the next example, we will evaluate the cube roots of some perfect cubes.

Example

Evaluate the following:

  1. 83
  2. 273
  3. 0

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? 83=2 because 222=8. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider (1)33=1.

We can also use factoring to simplify cube roots such as 1253. You can read this as “the third root of 125” or “the cube root of 125.” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let’s factor 125 and find that number.

Example

Simplify. 1253

The prime factors of 125 are 555, which can be rewritten as 53. The cube root of a cubed number is the number itself, so 533=5. You have found the cube root, the three identical factors that when multiplied together give 125. 125 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.

Here’s an example of how to simplify a radical that is not a perfect cube.

Example

Simplify. 32m53

In the example below, we use the idea that  (1)33=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

You could check your answer by performing the inverse operation. If you are right, when you cube 3xyx3 you should get 27x4y3.

(3xyx3)(3xyx3)(3xyx3)333xxxyyyx3x3x327x3y3x3327x3y3x27x4y3

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, and 21877, but you cannot simplify the radicals 100, 164, or 2,5006.

Let’s look at another example.

Example

Simplify. 24a53

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

Try It

In the same way that we learned earlier that we can estimate square roots, we can also estimate cube roots.

Example

Approximate 303 and also find its value using a calculator.

Try It

Nth Roots

We learned above that the cube root of a number is written with a small number 3, which looks like a3. This number placed just outside and above the radical symbol and is called the index. This little 3 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 a is a number that, when raised to the nth power, gives a. For example, 3 is the 5th root of 243 because (3)5=243. If a is a real number with at least one nth root, then the principal nth root of a is the number with the same sign as a that, when raised to the nth power, equals a.

The principal nth root of a is written as an, where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index of the radical.

Definition: Principal nth Root

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

Example

Evaluate each of the following:

  1. 325
  2. 814
  3. 18

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.
  • If n is even, xnn=|x|. (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.

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, and 21877 because the all have an odd numbered index, but you cannot evaluate the radicals 100, 164, or 2,5006 because they all have an even numbered index.

Try It

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 if n is odd, and xnn=|x| 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.