Roots and Radicals

# 72 Simplify and Use Square Roots

### Learning Objectives

By the end of this section, you will be able to:

- Simplify expressions with square roots
- Estimate square roots
- Approximate square roots
- Simplify variable expressions with square roots

Before you get started, take this readiness quiz.

### Simplify Expressions with Square Roots

Remember that when a number is multiplied by itself, we write and read it “n squared.” For example, reads as “15 squared,” and 225 is called the square of 15, since .

If , then is the square of .

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is is called a *square root* of .

If , then is a square root of .

Notice also, so is also a square root of 225. Therefore, both 15 and are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The *radical sign,* , denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because , . Notice that zero has only one square root.

is read as “the square root of .”

If , then , for .

The square root of , , is the positive number whose square is .

Since 15 is the positive square root of 225, we write . Fill in (Figure) to make a table of square roots you can refer to as you work this chapter.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, .

Simplify: ⓐ ⓑ ⓒ ⓓ .

ⓐ

ⓑ

ⓒ

ⓓ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

- ⓐ
- ⓑ

Simplify: ⓐ ⓑ .

ⓐ not a real number ⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ not a real number

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Simplify: ⓐ ⓑ .

ⓐ

ⓑ

Notice the different answers in parts ⓐ and ⓑ!

Simplify: ⓐ ⓑ .

ⓐ 7 ⓑ 5

Simplify: ⓐ ⓑ .

ⓐ 17 ⓑ

### Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at (Figure) below.

Number | Square Root |
---|---|

4 | = 2 |

5 | |

6 | |

7 | |

8 | |

9 | = 3 |

The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that must be between 2 and 3. Using inequality symbols, we write:

Estimate between two consecutive whole numbers.

Think of the perfect square numbers closest to 60. Make a small table of these perfect squares and their squares roots.

Locate 60 between two consecutive perfect squares. | |

is between their square roots. |

Estimate the square root between two consecutive whole numbers.

Estimate the square root between two consecutive whole numbers.

### Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the key on your calculator. You will use this key to approximate square roots.

When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately.’

Suppose your calculator has a 10-digit display. You would see that

If we wanted to round to two decimal places, we would say

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to 5, but are not exactly equal to 5.

Using the square root key on a calculator and then rounding to two decimal places, we can find:

Round to two decimal places.

Round to two decimal places.

Round to two decimal places.

### Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider . Can you think of an expression whose square is ?

When we use the radical sign to take the square root of a variable expression, we should specify that to make sure we get the *principal square root*.

However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write next to every radical.

What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6.

If we square , the exponent will become .

How does this help us take square roots? Let’s look at a few:

Simplify: ⓐ ⓑ .

ⓐ

ⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify: .

Simplify:

Simplify: .

Simplify: .

Access this online resource for additional instruction and practice with square roots.

### Key Concepts

- Note that the square root of a negative number is not a real number.
- Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
- We can estimate square roots using nearby perfect squares.
- We can approximate square roots using a calculator.
- When we use the radical sign to take the square root of a variable expression, we should specify that to make sure we get the principal square root.

#### Practice Makes Perfect

**Simplify Expressions with Square Roots**

In the following exercises, simplify.

6

8

3

10

not a real number

not a real number

5

7

**Estimate Square Roots**

In the following exercises, estimate each square root between two consecutive whole numbers.

**Approximate Square Roots**

In the following exercises, approximate each square root and round to two decimal places.

4.36

7.28

**Simplify Variable Expressions with Square Roots**

In the following exercises, simplify.

#### Everyday Math

**Decorating** Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

25 centimeters

**Decorating** Morris wants to have a square mosaic inlaid in his new patio. His budget allows for 2025 square inch tiles. How long can a side of the mosaic be?

#### Writing Exercises

Why is there no real number equal to ?

Answers will vary.

What is the difference between and ?

#### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Glossary

- square of a number

- If , then is the square of

- square root of a number

- If , then is a square root of

- square root notation

- If , then . We read as ‘the square root of .’