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**Find the two square roots of each number.**

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

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4-6 Learn to estimate square roots and solve problems using square roots. 2

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**Additional Example 1A: Estimating Square Roots of Numbers**

The square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 55? 55 72 = 49 49 < 55 82 = 64 64 > 55 55 is between 7 and 8 because 55 is between 49 and 64. 3

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**Additional Example 1B: Estimating Square Roots of Numbers Continued**

The square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 90? – 90 –92 = 81 81 < 90 –102 = 100 100 > 90 is between –9 and –10 because 90 is between 81 and 100. 90 – 4

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**Think: What are perfect squares close to 80? 80**

Check It Out: Example 1A The square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 80? 80 82 = 64 64 < 80 92 = 81 81 > 80 80 is between 8 and 9 because 80 is between 64 and 81. 5

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**The square root is between two integers. Name the integers.**

Check It Out: Example 1B The square root is between two integers. Name the integers. Think: What are perfect squares close to 45? – 45 –62 = 36 36 < 45 –72 = 49 49 > 45 is between –6 and –7 because 45 is between 36 and 49. 45 – 6

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**Additional Example 2: Application**

You want to sew a fringe on a square tablecloth with an area of 500 square inches. Calculate the length of each side of the tablecloth and the length of fringe you will need to the nearest tenth of an inch. The length of each side of the square is √500 . 484 < 500 < 529 List the perfect squares nearest 500. Find the square roots of the perfect squares. The number will be between 22 and 23. The length of each side of the table is about 22.4 in., and you will need about 89.6 in. of fringe.

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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 . 144 < 168 < 169 List the perfect squares nearest 168. < √168 √144 √169 12 < < 13 √168 Find the square roots of 144 and 169. 168 is closer to 169 than to 144. √168 13 Each side of the tent is about 13 feet long.

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**Additional Example 3: Approximating Square Roots to the Nearest Hundredth**

Approximate to the nearest hundredth. Step 1: Find the value of the whole number. 121 < 141 < 144 Find the perfect squares nearest 141. Find the square roots of the perfect squares. The number will be between 11 and 12. The whole number part of the answer is 11.

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**Additional Example 3 Continued**

Approximate to the nearest hundredth. Step 2: Find the value of the decimal. Find the difference between the given number, 141, and the lower perfect square. 141 – 121 = 20 Find the difference between the greater perfect square and the lower perfect square. 144 – 121 = 23 20 23 Write the difference as a ratio. Divide to find the approximate decimal value. 20 3 ≈ 0.869

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**Additional Example 3 Continued**

Approximate to the nearest hundredth. Step 3: Find the approximate value. Combine the whole number and decimal. = ≈ 11.87 Round to the nearest hundredth. The approximate value of to the nearest hundredth is

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Check It Out: Example 3 Approximate to the nearest hundredth. Step 1: Find the value of the whole number. 225 < 240 < 256 Find the perfect squares nearest 240. Find the square roots of the perfect squares. The number will be between 15 and 16. The whole number part of the answer is 15.

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**Check It Out: Example 3 Continued**

Approximate to the nearest hundredth. Step 2: Find the value of the decimal. Find the difference between the given number, 240, and the lower perfect square. 240 – 225 = 15 Find the difference between the greater perfect square and the lower perfect square. 256 – 225 = 31 15 31 Write the difference as a ratio. Divide to find the approximate decimal value. 15 31 ≈ 0.484

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**Check It Out: Example 3 Continued**

Approximate to the nearest hundredth. Step 3: Find the approximate value. Combine the whole number and decimal. = ≈ 15.48 Round to the nearest hundredth. The approximate value of to the nearest hundredth is

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**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. Using a calculator, ≈ …. Rounded, is 24.5. 15

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**Use a calculator to find 800. Round to the nearest tenth.**

Check It Out: Example 4 Use a calculator to find Round to the nearest tenth. Using a calculator, ≈ …. Rounded, is 28.3. 16

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**Lesson Quiz for Student Response Systems**

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 17

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**Each square root is between two integers. Name the two integers.**

Lesson Quiz Each square root is between two integers. Name the two integers. – 456 5 and 6 –22 and –21 Use a calculator to find each value. Round to the nearest tenth. 9.4 35.0 5. A square field has an area of 2000 square feet. To the nearest foot, how much fencing would be needed to enclose the field? 179 ft 18

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**Lesson Quiz for Student Response Systems**

1. Name the two integers the square root is between. A. 5 and 6 B. 6 and 7 C. 8 and 9 D. 38 and 40 19

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**Lesson Quiz for Student Response Systems**

2. Use a calculator to round to the nearest tenth. A. 2.3 B. 4.5 C. 6.7 D. 9.2 20 20

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**Lesson Quiz for Student Response Systems**

3. Name the two integers the square root is between. A. –2 and –3 B. –10 and –11 C. –13 and –14 D. –15 and –16 21 21

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