Finding Square Roots 3.9

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

Finding Square Roots Learn to estimate square roots to a given number of decimal places and solve problems using square roots.

A museum director wants to install a skylight to illuminate an unusual piece of art. It must be square and have an area of 300 square inches, with wood trim around it. Can you calculate the trim that you need? You can do this by using your knowledge of squares and square roots.

Example: Estimating Square Roots of Numbers Each square root is between two integers. Name the integers. Think: What are perfect squares close to 55? A. 55 72 = 49 49 < 55 82 = 64 64 > 55 55 is between 7 and 8. 7 < 55 < 8

Example: Estimating Square Roots of Numbers Continued Each square root is between two integers. Name the integers. Think: What are perfect squares close to 90? B. – 90 –92 = 81 81 < 90 –102 = 100 100 > 90 90 – is between –9 and –10. –9 < – 90 < –10

Try This Each square root is between two integers. Name the integers. Think: What are perfect squares close to 80? A. 80 82 = 64 64 < 80 92 = 81 81 > 80 is between 8 and 9. 80 8 < 80 < 9

Try This Each square root is between two integers. Name the integers. Think: What are perfect squares close to 45? B. – 45 –62 = 36 36 < 45 –72 = 49 49 > 45 45 – is between –6 and –7. –6 < – 45 < –7

Example 2: Problem Solving 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. Understand the Problem First find the length of a side. Then you can use the length of a side to find the perimeter, the length of fringe around the tablecloth.

Example 2 Continued Make a Plan The length of a side, in feet, is the number that you multiply by itself to get 500. To be accurate, find this number to the nearest tenth. If you do not know a step-by-step method for finding 500, use guess and check.

Example 2 Continued Solve Because 500 is between 222 and 232, the square root of 500 is between 22 and 23. Square root is between 22 and 22.5 Too high 22.52 = 506.25 Guess 22.5 1 Square root is between 22.2 and 22.5 Too low 22.22 = 492.84 Guess 22.2 2 Square root is between 22.3 and 22.4 Too low 22.32 = 497.29 Guess 22.3 4 Square root is between 22.2 and 22.4 Too high 22.42 = 501.76 Guess 22.4 3 22.0 22.2 22.4 22.6 The square root is between 22.3 and 22.4.

Example 2 Continued Solve The square root is between 22.3 and 22.4. To round to the nearest tenth, look at the next decimal place. Consider 22.35. 22.352 = 499.5225 Too low The square root must be greater than 22.35, so round up. To the nearest tenth, 500 is about 22.4. The length of each side of the table is about 22.4 in.

Example 2 Continued Solve The length of a side of the tablecloth is 22.4 inches, to the nearest tenth of an inch. Now estimate the length around the tablecloth. 4 • 22.4 = 89.6 Perimeter 4 • side You will need about 89.6 inches of fringe.

Example 2 Continued Look Back The length 90 inches divided by 4 is 22.5 inches. A 22.5-inch square has an area of 506.25 square inches, which is close to 500, so the answers are reasonable.

Try This You want to build a fence around a square garden that is 250 square feet. Calculate the length of one side of the garden and the total length of the fence, to the nearest tenth. Understand the Problem First find the length of a side. Then you can use the length of a side to find the perimeter, the length of the fence.

Try This Make a Plan The length of a side, in feet, is the number that you multiply by itself to get 250. To be accurate, find this number to the nearest tenth. If you do not know a step-by-step method for finding 250, use guess and check.

Try This Solve Because 250 is between 152 and 162, the square root of 250 is between 15 and 16. Guess 15.5 15.52 = 240.25 Too Low Square root is between 15.5 and 16 Guess 15.9 15.92 = 252.81 Too high Square root is between 15.5 and 15.9 Guess 15.7 15.72 = 246.49 Too Low Square root is between 15.7 and 15.9 Guess 15.8 15.82 = 249.46 Too Low Square root is between 15.8 and 15.9 1 3 4 2 15.5 15.7 15.9 16.1 The square root is between 15.8 and 15.9.

Try This Solve To round to the nearest tenth, look at the next decimal place. Consider 15.85. 15.852 = 251.2225 The square root is lower than 15.85, so round down. To the nearest tenth, 250 is about 15.8. The length of each side of the garden is about 15.8 ft.

Try This Solve The length of a side of the garden is 15.8 feet, to the nearest tenth of a foot. Now estimate the length around the garden. 4 • 15.8 = 63.2 Perimeter 4 • side You will need about 63.2 feet of fence.

Try This Look Back The length 63.2 feet divided by 4 is 15.8 feet. A 15.8 foot square has an area of 249.64 square feet, which is close to 250, so the answers are reasonable.

Examples 3: Using a Calculator to Estimate the Value of a Square Root Use a calculator to find 500. Round to the nearest tenth. Using a calculator, 500 ≈ 22.36067977…. Rounded, 500 is 22.4.

Try This Use a calculator to find 200. Round to the nearest tenth. Using a calculator, 200 ≈ 14.14213562…. Rounded, 200 is 14.1.

Lesson Quiz Each square root is between two integers. Name the two integers. 1. 27 2. – 456 5 and 6 –22 and –21 Use a calculator to find each value. Round to the nearest tenth. 3. 89 4. 1223 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