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 Cune

We know that $5^2=25, \text{ and }\sqrt{25}=5$, but what if we want to “undo” $5^3=125, \text{ or }5^4=625$? 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 $10th$ 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 $\sqrt[3]{{}}$. This little $3$ distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.

Caution! Be careful to distinguish between $\sqrt[3]{x}$, the cube root of x, and $3\sqrt{x}$, 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 ${a}^{3}=8$. We want to find what number raised to the $3$rd power is equal to $8$. Since ${2}^{3}=8$, 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. $\sqrt[3]{-8}$
2. $\sqrt[3]{27}$
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 $−8$? $\sqrt[3]{-8}=-2$ because $-2\cdot -2\cdot -2=-8$. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider $\sqrt[3]{{{(-1)}^{3}}}=-1$.

We can also use factoring to simplify cube roots such as $\sqrt[3]{125}$. 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. $\sqrt[3]{125}$

The prime factors of $125$ are $5\cdot5\cdot5$, which can be rewritten as $5^{3}$. The cube root of a cubed number is the number itself, so $\sqrt[3]{{{5}^{3}}}=5$. 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. $\sqrt[3]{32{{m}^{5}}}$

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

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

$\begin{array}{l}\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\\-3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\\-27\cdot {{x}^{3}}\cdot {{y}^{3}}\cdot \sqrt[3]{{{x}^{3}}}\\-27{{x}^{3}}{{y}^{3}}\cdot x\\-27{{x}^{4}}{{y}^{3}}\end{array}$

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 $\sqrt[3]{-81},\ \sqrt[5]{-64}$, and $\sqrt[7]{-2187}$, but you cannot simplify the radicals $\sqrt[{}]{-100},\ \sqrt[4]{-16}$, or $\sqrt[6]{-2,500}$.

Let’s look at another example.

Example

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

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

Nth Roots

We learned above that the cube root of a number is written with a small number $3$, which looks like $\sqrt[3]{a}$. 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 ${\left(3\right)}^{5}=243$. 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 $\sqrt[n]{a}$, 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 $\sqrt[n]{a}$, 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. $\sqrt[5]{-32}$
2. $\sqrt[4]{81}$
3. $\sqrt[8]{-1}$

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

When working with exponents and radicals:

• If n is odd, $\sqrt[n]{{{x}^{n}}}=x$.
• If n is even, $\sqrt[n]{{{x}^{n}}}=\left| x \right|$. (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.

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 $\sqrt[3]{-81},\ \sqrt[5]{-64}$, and $\sqrt[7]{-2187}$ because the all have an odd numbered index, but you cannot evaluate the radicals $\sqrt[{}]{-100},\ \sqrt[4]{-16}$, or $\sqrt[6]{-2,500}$ because they all have an even numbered index.

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 $\sqrt[n]{{{x}^{n}}}=x$ if n is odd, and $\sqrt[n]{{{x}^{n}}}=\left| x \right|$ 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.