9.1 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
Be Prepared 9.1
Before you get started, take this readiness quiz.
Simplify: ⓐ 9 2 ⓑ (−9)2 ⓒ −9 2 .
If you missed this problem, review Example 1.50.
Be Prepared 9.2
Round 3.846 to the nearest hundredth.
If you missed this problem, review Example 1.94.
Be Prepared 9.3
For each number, identify whether it is a real number or not a real number:
ⓐ√ −100 ⓑ √−100.
If you missed this problem, review Example 1.113.
Simplify Expressions with Square Roots
Remember that when a number n is multiplied by itself, we write n 2 and read it “n squared.” For example, 15 2 reads as “15 squared,” and 225 is called the square of 15, since 152=225.
Square of a Number
If n 2 =m, then m is the square of n.
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 m is called a square root of m.
Square Root of a Number
If n2=m, then n is a square root of m
Notice (−15)2=225 also, so −15 is also a square root of 225. Therefore, both 15 and −15 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, √m, 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 02=0, √0=0. Notice that zero has only one square root.
Square Root Notation
Since 15 is the positive square root of 225, we write √225=15. Fill in Figure 9.2 to make a table of square roots you can refer to as you work this chapter.
Example 9.1
Simplify: ⓐ √36 ⓑ√196 ⓒ √−81 ⓓ √−289.
Solution
ⓐ
| √36 | |
| Since 62=36 | 6 |
ⓑ
| √196 | |
| Since 142=196 | 14 |
ⓒ
| √−81 | |
| The negative is in front of the radical sign. | −9 |
ⓓ
| √−289 | |
| The negative is in front of the radical sign. | −17 |
Try It 9.1
Simplify: ⓐ √−49 ⓑ √225.
Try It 9.2
Simplify: ⓐ √64 ⓑ √−121.
Example 9.2
Simplify: ⓐ√−169 ⓑ √ −64.
Solution
ⓐ
| √−169 | |
| There is no real number whose square is −169. | √−169 is not a real number. |
ⓑ
| √−64 | |
| The negative is in front of the radical. | −8 |
Try It 9.3
Simplify: ⓐ √−196 ⓑ √−81 .
Try It 9.4
Simplify: ⓐ √−49 ⓑ √−121
When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.
Example 9.3
Simplify: ⓐ √25 +√144 ⓑ√ 25+144
Solution
ⓐ
| √ 25 + √ 144 | |
| Use the order of operations. | 5+12 |
| Simplify. | 17 |
ⓑ
| √ 25+144 | |
| Simplify under the radical sign. | √169 |
| Simplify. | 13 |
| Notice the different answers in parts ⓐ and ⓑ! |
Try It 9.5
Simplify: ⓐ √9 +√16 ⓑ √9+16
Try It 9.6
Simplify: ⓐ √64+225 ⓑ √64+√225
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 Table 9.1 below.
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 5–√ must be between 2 and 3. Using inequality symbols, we write:
Example 9.4
Estimate √60 between two consecutive whole numbers.
Solution
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. | |
| 60−−√ is between their square roots. |
Try It 9.7
Estimate the square root √38 between two consecutive whole numbers.
Try It 9.8
Estimate the square root √84 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 √x 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
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:
Example 9.5
Round √17to two decimal places.
Solution
| √ 17 | |
| Use the calculator square root key. | 4.123105626… |
| Round to two decimal places. | 4.12 |
| √ 17 ≈4.12 |
Try It 9.9
Round √11 to two decimal places.
Try It 9.10
Round √13 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 √9x2. Can you think of an expression whose square is 9x2?
When we use the radical sign to take the square root of a variable expression, we should specify that x≥0 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 x≥0 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.
Media
Access this online resource for additional instruction and practice with square roots.
- Square Roots
Section 9.1 Exercises
Practice Makes Perfect
Simplify Expressions with Square Roots
In the following exercises, simplify.
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.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.
Everyday Math
49.
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? 50.
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
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?