This document provides examples for estimating square roots: 1. It shows how to find the two integers between which a square root lies and use perfect squares to estimate values. 2. Examples estimate the side length of squares given the total area and show approximating square roots to the nearest hundredth. 3. Using a calculator to estimate square roots and rounding to the nearest tenth is demonstrated.

1. Warm Up California Standards Lesson Presentation Preview

2. Warm Up Find the two square roots of each number. Evaluate each expression.  12  16 20 119 1. 144 2. 256 3. 8 + 144 4. 7 289

3. 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. California Standards

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

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

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

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

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

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

10. Additional Example 3 Continued 11 + 0.609 = 11.609 Combine the whole number and decimal. Step 3 Find the approximate value. 11.609 ≈ 11.61 Round to the nearest hundredth. The approximate value of 135 to the nearest hundredth is 11.61. Approximate √135 to the nearest hundredth.

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

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

13. Check It Out! Example 3 Continued 13 + 0.407 = 13.407 Combine the whole number and decimal. Step 3 Find the approximate value. 13.407 ≈ 13.41 Round to the nearest hundredth. The approximate value of 180 to the nearest hundredth is 13.41. Approximate √180 to the nearest hundredth.

14. Additional Example 4: Using a Calculator to Estimate the Value of a Square Root Use a calculator to find 600. Round to the nearest tenth. 600 ≈ 24.5 Use a calculator. Round to the nearest tenth. 600 rounded to the nearest tenth is 24.5. 600 ≈ 24.494897…

15. Check It Out! Example 4 800 ≈ 28.3 Use a calculator to find 800. Round to the nearest tenth. Use a calculator. Round to the nearest tenth. 800 rounded to the nearest tenth is 28.3. 800 ≈ 28.28427125…

16. Lesson Quiz Each square root is between two integers. Name the integers. 1. 27 2. 456 3. Approximate 189 to the nearest hundredth. 4. Use a calculator to find 1219. Round to the nearest tenth. 5. A square room requires 154 ft 2 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. 5 and 6 21 and 22 13.75 34.9 12 ft