# 7.1: Defining nth Roots

**At Grade**Created by: CK-12

**Practice**nth Roots

The volume of a cube is found to be

### Guidance

So far, we have seen exponents with integers and the square root. In this concept, we will link roots and exponents. First, let’s define additional roots. Just like the square and the square root are inverses of each other, the inverse of a cube is the cubed root. The inverse of the fourth power is the fourth root.

The **root** of a number,

#### Example A

Find

**Solution:** To simplify a number to the sixth root, there must be 6 of the same factor to pull out of the root.

Therefore,

From this example, we can see that it does not matter where the exponent is placed, it will always cancel out with the root.

So, it does not matter if you evaluate the root first or the exponent.

**The nth Root Theorem:** For any real number

#### Example B

Evaluate without a calculator:

a)

b)

**Solution:**

a) If you solve this problem as written, you would first find

However, this would be very difficult to do without a calculator. This is an example where it would be easier to apply the root and then the exponent. Let’s rewrite the expression and solve.

b) This problem does not need to be rewritten.

#### Example C

Simplify:

a)

b)

**Solution:**

a) To simplify the fourth root of a number, there must be 4 of the same factor to pull it out of the root. Let’s write the prime factorization of 64 and simplify.

Notice that there are 6 2’s in 64. We can pull out 4 of them and 2 2’s are left under the radical.

b) Just like simplifying fractions with square roots, we can separate the numerator and denominator.

Notice that because the

**Intro Problem Revisit** Recall that the volume of a cube is *s* is the length of each side. So to find the side length, take the cube root of

First, you can separate this number into two different roots,

Therefore, the length of the cube's side is

### Guided Practice

Simplify each expression below, without a calculator.

1.

2.

3.

4.

#### Answers

1. First, you can separate this number into two different roots,

When looking at the

2.

3. Write out 9216 in the prime factorization and place factors into groups of 5.

4. Reduce the fraction, separate the numerator and denominator and simplify.

In the red step, we rationalized the denominator by multiplying the top and bottom by

### Explore More

Reduce the following radical expressions.

- \begin{align*}\sqrt[3]{81}\end{align*}
81−−√3 - \begin{align*}\sqrt[4]{625}^3\end{align*}
- \begin{align*}\sqrt{9^5}\end{align*}
- \begin{align*}\sqrt[5]{128}\end{align*}
- \begin{align*}\sqrt{\sqrt{10000}}\end{align*}
- \begin{align*}\sqrt{\frac{25}{8}}^4\end{align*}
- \begin{align*}\sqrt[6]{64}^5\end{align*}
- \begin{align*}\sqrt[3]{\frac{8}{81}}^2\end{align*}
- \begin{align*}\sqrt[4]{\frac{243}{16}}\end{align*}
- \begin{align*}\sqrt[3]{24x^5}\end{align*}
- \begin{align*}\sqrt[4]{48x^7y^{13}}\end{align*}
- \begin{align*}\sqrt[5]{\frac{160x^8}{y^7}}\end{align*}
- \begin{align*}\sqrt[3]{1000x^6}^2\end{align*}
- \begin{align*}\sqrt[4]{\frac{162x^5}{y^3z^{10}}}\end{align*}
- \begin{align*}\sqrt{40x^3y^4}^3\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.1.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to define and use @$\begin{align*}n^{th}\end{align*}@$ roots.

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## Date Created:

Mar 12, 2013## Last Modified:

Jun 04, 2015## Vocabulary

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