3-6 Estimating Square Roots

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Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

3-6 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

3-6 Estimating Square Roots

Problem of the DayA pyramid of blocks is built in layers. The bottom layer has 62, or 36, blocks. The next layer has 52 blocks, and so on until the top layer has 1 block. How many blocks are there in all?91 blocks

3-6 Estimating Square Roots

Learn to estimate square roots and solve problems using square roots.

3-6

3-6 Estimating Square Roots

Additional Example 1A: Estimating Square Roots of Numbers

The square root is between two integers. Name the integers. Explain your answer.

55Think: What are perfect squares close to 55?

72 = 49 49 < 55

82 = 64 64 > 55

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

3-6 Estimating Square Roots

90–

Additional Example 1B: Estimating Square Roots of Numbers Continued

Think: What are perfect squares close to 90?

–92 = 81 81 < 90

–102 = 100 100 > 90

The square root is between two integers. Name the integers. Explain your answer.

is between –9 and –10 because 90 is between 81 and 100.

90–

3-6 Estimating Square Roots

Check It Out: Example 1A

The square root is between two integers. Name the integers. Explain your answer.

80Think: What are perfect squares close to 80?

82 = 64 64 < 80

92 = 81 81 > 80

80 is between 8 and 9 because 80 is between 64 and 81.

3-6 Estimating Square Roots

45–

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

Think: What are perfect squares close to 45?

(–6)2 = 36 36 < 45

(–7)2 = 49 49 > 45

Check It Out: Example 1B

is between –6 and –7 because 45 is between 36 and 49.

45–

3-6 Estimating Square Roots

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.

Additional Example 2: Application

484 < 500 < 529 List the perfect squares nearest 500.

Find the square roots of the perfect squares.

The length of each side of the table is about 22.4 in., and you will need about 89.6 in. of fringe.

The number will be between 22 and 23.

The length of each side of the square is √500 .

3-6 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 List the perfect squares nearest 168.

Find the square roots of 144 and 169.

< < √168√144 √169

12 < < 13√168

√168 13

Each side of the tent is about 13 feet long.

168 is closer to 169 than to 144.

3-6 Estimating Square Roots

Approximate to the nearest hundredth.

Additional Example 3: Approximating Square Roots to the Nearest Hundredth

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.

Step 1: Find the value of the whole number.

3-6 Estimating Square Roots

Approximate to the nearest hundredth.

Additional Example 3 Continued

141 – 121 = 20Find the difference between the givennumber, 141, and the lower perfect square.

Find the difference between the greater perfect square and the lowerperfect square.

Write the difference as a ratio.

20 3 ≈ 0.869

Step 2: Find the value of the decimal.

144 – 121 = 23

20 23

Divide to find the approximate decimal value.

3-6 Estimating Square Roots

11 + 0.869 = 11.869Combine the whole number anddecimal.

Round to the nearest hundredth.

Step 3: Find the approximate value.

11.869 ≈ 11.87

Approximate to the nearest hundredth.

Additional Example 3 Continued

The approximate value of to the nearesthundredth is 11.87.

3-6 Estimating Square Roots

Approximate to the nearest hundredth.

Check It Out: Example 3

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.

Step 1: Find the value of the whole number.

3-6 Estimating Square Roots

Check It Out: Example 3 Continued

240 – 225 = 15Find the difference between the givennumber, 240, and the lower perfect square.

Find the difference between the greater perfect square and the lowerperfect square.

Write the difference as a ratio.

15 31 ≈ 0.484

Step 2: Find the value of the decimal.

256 – 225 = 31

15 31

Divide to find the approximate decimal value.

Approximate to the nearest hundredth.

3-6 Estimating Square Roots

15 + 0.484 = 15.484Combine the whole number anddecimal.

Round to the nearest hundredth.

Step 3: Find the approximate value.

15.484 ≈ 15.48

Check It Out: Example 3 Continued

Approximate to the nearest hundredth.

The approximate value of to the nearesthundredth is 15.48.

3-6 Estimating Square Roots

Additional Example 4: Using a Calculator to Estimate the Value of a Square Root

Using a calculator, 600 ≈ 24.49489742….

Rounded, 600 is 24.5.

Use a calculator to find 600. Round to the nearest tenth.

3-6 Estimating Square Roots

Check It Out: Example 4

Rounded, 800 is 28.3.

Using a calculator, 800 ≈ 28.2842712….

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

3-6 Estimating Square Roots

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

3-6 Estimating Square Roots

Lesson Quiz

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

1. 27 2. – 456

Use a calculator to find each value. Round to the nearest tenth.

3. 89 4. 1223

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?

5 and 6 –22 and –21

9.4 35.0

179 ft

3-6 Estimating Square Roots

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

Lesson Quiz for Student Response Systems

3-6 Estimating Square Roots

2. Use a calculator to round to the nearest tenth.

A. 2.3

B. 4.5

C. 6.7

D. 9.2

Lesson Quiz for Student Response Systems

3-6 Estimating Square Roots

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

Lesson Quiz for Student Response Systems