Evaluating Algebraic Expressions

4-7 Estimating Square Roots

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Evaluating Algebraic Expressions

4-7 Estimating Square Roots

Warm UpFind the two square roots of each number.

Evaluate each expression.

12 16

20 119

1. 144 2. 256

3. 8 + 144 4. 7 289

Evaluating Algebraic Expressions

4-7 Estimating Square Roots

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

Evaluating Algebraic Expressions

4-7 Estimating Square RootsAdditional Example 1: Estimating Square Roots of

NumbersThe 55 is between two integers. Name the integers. Explain your answer.

List perfect squares near 55.36, 49, 64, 8149 < 55 < 64

Find the square roots of the perfect squares.

55

55 is between 7 and 8 because 55 is between 49 and 64.

Find the perfect squares nearest 55.

49 < 55 < 64

7 < 55 < 8

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck It Out! Example 1

List perfect squares near 80.49, 64, 81, 10064 < 80 < 81

Find the square roots of the perfect squares.

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.

Find the perfect squares nearest 80.

64 < 80 < 81

8 < 80 < 9

Evaluating Algebraic Expressions

4-7 Estimating Square RootsA 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.

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.

The length of each side of the square is √185 .

< < √185√169 √196

13 < < 14√185

√185 14

144, 169, 196, 225 List perfect squares near 185.

185 is closer to 196 than to 169, so 185 is closer to 14 than 13.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck 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 Find the perfect squares nearest 168.

Find the square roots of the perfect squares.

< < √168√144 √16912 < < 13√168

√168 13Each side of the tent is about 13 feet long.

121, 144, 169, 196 List perfect squares near 168.

168 is closer to 169 than to 144, so 168 is closer to 13 than 12.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsAdditional 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.

√121 < √135 < 144√

11 < 135 < 12√

Step 1 Find the value of the whole number.

The number will be between 11 and 12.

Approximate √135 to the nearest hundredth.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsAdditional Example 3 Continued

135 – 121 = 14

14 ÷ 23 ≈ 0.609

Find the difference between the given number, 135, and the lower perfect square.

Approximate √135 to the nearest hundredth.Step 2 Find the value of the decimal.

Write the difference as a ratio.

144 – 121 = 23Find the difference between the greater perfect square and the lower perfect square.

1423

Divide to find the approximate decimal value.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsAdditional 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.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck 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.

√169 < √180 < 196√

13 < 180 < 14√

Step 1 Find the value of the whole number.

The number will be between 13 and 14.

Approximate √180 to the nearest hundredth.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck 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 = 27Find the difference between the greater perfect square and the lower perfect square.

1127

Divide to find the approximate decimal value.

Approximate √180 to the nearest hundredth.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck 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.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsAdditional Example 4: Using a Calculator to Estimate the

Value of a Square RootUse a calculator to find 600. Round to the nearest tenth.

600 ≈ 24.5

600 ≈ 24.494897… Use a calculator.

Round to the nearest tenth.

600 rounded to the nearest tenth is 24.5.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsCheck It Out! Example 4

800 ≈ 28.3

800 ≈ 28.28427125…

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.

Evaluating Algebraic Expressions

4-7 Estimating Square RootsLesson 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 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.

5 and 6 21 and 22

13.75

34.9

12 ft