1. Preview
Warm Up
California Standards
Lesson Presentation

2. Find the two square roots of each number.
Warm Up
Find the two square roots of each number.
Evaluate each expression.
±12
±16 20
119

3. California
Standards
NS2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not a square, determine without a calculator the two integers between which its square root lies and explain why.

4. Additional Example 1: Estimating Square Roots of Numbers
The 55 is between two integers. Name the integers. Explain your answer. 55
36, 49, 64, 81
List perfect squares near 55.
49 < 55 < 64
Find the perfect squares nearest 55.
49 < 55 < 64
Find the square roots of the perfect squares.
7 < 55 < 8
55 is between 7 and 8 because 55 is between 49 and 64.

5. Check It Out! Example 1
The 80 is between two integers. Name the integers. Explain your answer. 80
49, 64, 81, 100
List perfect squares near 80.
64 < 80 < 81
Find the perfect squares nearest 80.
64 < 80 < 81
Find the square roots of the perfect squares.
8 < 80 < 9
80 is between 8 and 9 because 80 is between 64 and 81.

6. Additional Example 2: Recreation Application
A Coast Guard boat searching for a lost sailboat covers a square area of 185 mi2. What is the approximate length of each side of the square area? Round your answer to the nearest mile.
The length of each side of the square is
√185 .
144, 169, 196, 225
List perfect squares near 185.
169 < 185 < 196
Find the perfect squares nearest 185.
Find the square roots of the perfect squares.
<
√185
√169
√196
13 < < 14
√185
185 is closer to 196 than to 169, so is closer to 14 than 13.
√185  14
Each side of the search area is about 14 miles long.

7. Check It Out! Example 2
A tent was advertised in the newspaper as having an enclosed square area of 168 ft2. What is the approximate length of the sides of the square area? Round your answer to the nearest foot.
The length of each side of the square is
√168 .
121, 144, 169, 196
List perfect squares near 168.
144 < 168 < 169
Find the perfect squares nearest 168.
<
√168
√144
√169
Find the square roots of the perfect squares.
12 < < 13
√168
168 is closer to 169 than to 144, so is closer to 13 than 12.
√168  13
Each side of the tent is about 13 feet long.

8. Additional Example 3: Approximating Square Roots to the Nearest Hundredth
Approximate √135 to the nearest hundredth.
Step 1 Find the value of the whole number.
121 < 135 < 144
Find the perfect squares nearest 135.
√121 <
√135 <
144 √
Find the square roots of the perfect squares.
11 <
135
< 12 √
The number will be between 11 and 12.
The whole number part of the answer is 11.

9. Additional Example 3 Continued
Approximate √135 to the nearest hundredth.
Step 2 Find the value of the decimal.
Find the difference between the given number, 135, and the lower perfect square.
135 – 121 = 14
Find the difference between the greater perfect square and the lower perfect square.
144 – 121 = 23
1423
Write the difference as a ratio.
Divide to find the approximate decimal value.
14 ÷ 23 ≈ 0.609

10. Additional Example 3 Continued
Approximate √135 to the nearest hundredth.
Step 3 Find the approximate value.
Combine the whole number and decimal. =
≈ 11.61
Round to the nearest hundredth.
The approximate value of to the nearest hundredth is

11. Check It Out! Example 3
Approximate √180 to the nearest hundredth.
Step 1 Find the value of the whole number.
169 < 180 < 196
Find the perfect squares nearest 180.
√169 <
√180 <
196 √
Find the square roots of the perfect squares.
13 <
180
< 14 √
The number will be between 13 and 14.
The whole number part of the answer is 13.

12. Check It Out! Example 3 Continued
Approximate √180 to the nearest hundredth.
Step 2 Find the value of the decimal.
Find the difference between the given number, 180, and the lower perfect square.
180 – 169 = 11
Find the difference between the greater perfect square and the lower perfect square.
196 – 169 = 27
1127
Write the difference as a ratio.
Divide to find the approximate decimal value.
11 ÷ 27 ≈ 0.407

13. Check It Out! Example 3 Continued
Approximate √180 to the nearest hundredth.
Step 3 Find the approximate value.
Combine the whole number and decimal. =
≈ 13.41
Round to the nearest hundredth.
The approximate value of to the nearest hundredth is

14. Use a calculator to find 600. Round to the nearest tenth.
Additional Example 4: Using a Calculator to Estimate the Value of a Square Root
Use a calculator to find Round to the nearest tenth.
600 ≈ …
Use a calculator.
600 ≈ 24.5
Round to the nearest tenth.
600 rounded to the nearest tenth is 24.5.

15. Check It Out! Example 4
Use a calculator to find Round to the nearest tenth.
800 ≈ …
Use a calculator.
800 ≈ 28.3
Round to the nearest tenth.
800 rounded to the nearest tenth is 28.3.

16. Lesson Quiz
Each square root is between two integers. Name the integers.
5 and 6
21 and 22
3. Approximate to the nearest hundredth.
4. Use a calculator to find Round to the nearest tenth.
13.75
34.9
5. A square room requires 154 ft2 of wall-to-wall carpeting to cover the floor. What is the length of each side of the room? Round your answer to the nearest foot.
12 ft