Square Roots and Cube Roots

Develop Skills and Strategies Part 1: Introduction Square Roots and Cube Roots MAFS 8.EE.1.2 In Lesson 1 you learned the properties of integer exponents. Now, take a look at this problem. The length of each side of a square measures s inches long. The area of the square is 49 in. 2 What is the length of one side of the square? s s s s Explore It Use the math you know to answer the question. Describe in words how to find the area of the square given that each side is s inches long. Write a multiplication expression using the variable s to represent the area of the square. Write an expression using the variable s and an exponent to represent the area of the square. Write an equation setting your expression equal to the area of the square given in the problem. Consider the factors of 49. Explain what the two sides of the equation have in common when you write each as the product of two factors. 12

Part 1: Introduction Find Out More The number 49 is one of a set of numbers called perfect squares. A perfect square is a number that results from multiplying an integer by itself. The first 15 square numbers are shown. 1 2 5 1 4 2 5 16 7 2 5 49 10 2 5 100 13 2 5 169 2 2 5 4 5 2 5 25 8 2 5 64 11 2 5 121 14 2 5 196 3 2 5 9 6 2 5 36 9 2 5 81 12 2 5 144 15 2 5 225 Look at the equation you wrote on the previous page, s 2 = 49. How do you solve an equation where a variable squared is equivalent to a perfect square? You have solved equations before by using inverse operations. You solved addition equations by subtracting. You solved division equations by multiplying. What is the inverse operation of squaring a number? The inverse operation of squaring is finding the square root. A square root of a number is any number that you can multiply by itself to get your original number. For example, 3 is a square root of 9, because 3 3 = 9. Another square root of 9 is 23, because (23) (23) 5 9. The symbol Ï means positive square root. So, Ï 9 5 3. s 2 5 49 Ï s 2 5 Ï 49 Ï s 2 5 Ï 7 2 s 5 7 The inverse of squaring is finding a square root. Find the square root of both sides. 49 is a perfect square. The length of one side of the square is 7 inches. Reflect 1 What is the difference between dividing 16 by 2 and finding the square roots of 16? 13

Part 2: Modeled Instruction Read the problem below. Then explore how to solve equations with cubes and cube roots. Each edge of a cube measures a feet long. The volume of the cube is 125 ft 3. What is the measure of each edge of the cube? Picture It Draw and label the cube. a a a Volume 5125 ft 3 The length, width, and height of the cube each measure a feet. Solve It You can apply the formula for the volume of a cube. The volume of the cube is the product of its length, width, and height. a a a 5 V length 5 a, width 5 a, and height 5 a a 3 5 V Substitute the given volume of the cube for V. a 3 5 125 You can use this equation to find the value of a. 14

Part 2: Guided Instruction Connect It Now you will solve the problem from the previous page. 2 Complete the prime factorization of 125. 125 25 3 Write 125 as the product of three factors. 4 Write 125 as a power of base 5. 5 What does 125 have in common with a 3 when 125 is written as a power? The product of an integer multiplied together three times is a perfect cube. Finding the cube root is the inverse of cubing a number. The cube root of a number is the number that is multiplied together three times to produce the original number. The symbol 3 Ï means find the cube root. 6 Look at Solve It on the previous page. The equation shows a variable cubed equal to a perfect cube. Use the cube root to complete the solution. a 3 5 125 Ï 3 a 3 5 Ï 3 1250 Ï 3 a 3 5 Ï 3 533 Try It Solution: Each edge of the cube is feet long. 5 Use what you just learned to solve these problems. Show your work on a separate sheet of paper. 7 Solve: y 3 5 8 8 Solve: x 3 5 27 15

Part 3: Modeled Instruction Read the problem below. Then explore how to use square roots and cube roots to solve word problems. City Park is a square piece of land with an area of 10,000 square yards. What is the length of the fence that encloses the park? Picture It You can draw a diagram to help solve the problem. The park is a square. The fence runs along the outside edge of the park. Area 510,000 yd 2 City park Fence The length of the fence is the perimeter of the square. Solve It To find the perimeter of the square park, you need to know the length of one side of the square. Let f be the length of one side of the square. A 5 10,000 Area of the park is 10,000 yd 2 f 2 5 10,000 Area equals the length of one side squared. 16

Part 3: Guided Instruction Connect It Now you will solve the problem from the previous page. 9 What number squared equals 10,000? 10 Look at Solve It on the previous page. Solve the equation for f. f 2 5 10,000 11 What is the length of each side of the park? 12 Write and solve an equation to find the perimeter of the park. 13 What is the length of the fence that encloses the park? 14 The park s rectangular garden area is 450 square yards. Its length is twice its width. Find the dimensions of the garden. Begin with the equation (2w)(w) 450. Rewrite the equation using exponents. Divide both sides by 2. Solve and write the garden s dimensions. Try It Use what you just learned about square roots and cube roots to solve these problems. 15 The volume of a cube is 1,000 cm 3. What is the length of an edge? 16 A gift box in the shape of a cube has a volume of 216 cm 3. What is the area of the base of the box? 17 A scientist finds the temperature of a sample at the beginning of an experiment is t C. After 1 hour, the temperature is t 2 C. If the temperature after 1 hour is 81 C, what are two possible original temperatures? What is the difference between the possible original temperatures? 17

Part 4: Guided Practice Study the student model below. Then solve problems 18 20. In this problem, you will divide before you find the square root. Student Model The distance in feet that a freely falling dropped object falls in t seconds is given by the equation d 16 5 t2. How long does it take a dropped object to fall 64 feet? Look at how you could solve this problem. The given equation is: d 16 5 t2 Substitute 64 for d: 64 16 5 t2 Simplify: 4 5 t 2 Take the square root of both sides: Ï 4 5 Ï t 2 Pair/Share How far does an object fall in 1 second? Solution: t 5 2 The object takes 2 seconds to fall 64 feet. What information do you need to calculate the volume of a cube? 18 The area of the top face of a cube is 9 square meters. What is the volume of the cube? Show your work. Pair/Share The cube has 6 faces. What does the expression 6 9 describe? Solution: 18

Part 4: Guided Practice 19 The length of each side of a cube is x centimeters. If x is an integer, why can t the volume of the cube equal 15 cm 3? Show your work. Write an equation showing a variable expression for volume is equal to 15. Solution: Pair/Share Are all perfect cubes also multiples of 3? Are all multiples of 3 also perfect cubes? Discuss. 20 Yesterday, there were b milligrams of bacteria in a lab experiment. Today, there are b 2 milligrams of bacteria. If there are 400 milligrams today, how many milligrams of bacteria were there yesterday? A 20 milligrams Do you square a number or find the square root to solve the problem? B C 200 milligrams 1,600 milligrams D 160,000 milligrams Eva chose B as the correct answer. How did she get that answer? Pair/Share Talk about the problem and then write your answer together. 19

Part 5: MAFS Practice Solve the problems. 1 Solve a 3 5 64. A a 5 4 B a 5 8 C a 5 21 D a 5 32 2 Which number is a perfect square? A 8 B 18 C 200 D 225 3 The fractions below are the values of x in the given equations. Write the correct fraction inside the box for each equation. A x 2 5 4 } 9 B x 3 5 27 }} 64 C x 2 5 81 }} 64 D x 3 5 1 } 8 9 }} 8 1 } 2 3 } 4 2 } 3 20

Part 5: MAFS Practice 4 Use the numbers shown to make the two equations true. Each number can be used only once. Write the number in the appropriate box for each equation. 3 6 100 36 1,000 1,000,000 Ï 5 3 Ï 5 5 If x is a positive integer, is Ï 1 greater than, less than, or equivalent to Ï 3 1? x 2 x 3 Show your work. Answer 6 Describe how you could use inverse operations to solve the equation Ï x 5 4. Self Check Go back and see what you can check off on the Self Check on page 1. 21

NAME DATE PERIOD Lesson 8 Reteach Roots A square root of a number is one of its two equal factors. A radical sign, is used to indicate a positive square root. Every positive number has both a negative and positive square root. Examples Find each square root. 1. 1 Find the positive square root of 1; 1 2 = 1, so 1 = 1. 2. - 16 Find the negative square root of 16; (-4) 2 = 16, so - 16 = -4. 3. ± 0.25 Find both square roots of 0.25; 0.5 2 = 0.25, so ± 0.25 = ±0.5. 4. -49 There is no real square root because no number times itself is equal to -49. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Example 5 Solve a 2 = 4. Check your solution(s). 9 a 2 = 4 9 a = ± 4 9 a = 2 3 or - 2 3 Write the equation. Defi nition of square root Check 2 2 3 3 = 4 9 and (- 2 3) (- 2 = 3) 4 9. The equation has two solutions, 2 3 and - 2 3. Exercises Find each square root. 1. 4 2. 9 3. - 49 4. - 25 5. ± 0.01 6. - 0.64 7. 9 16 8. -1 25 ALGEBRA Solve each equation. Check your solution(s). 9. x 2 = 121 10. a 2 = 3,600 11. p 2 = 81 100 12. t 2 = 121 196 Course 3 Chapter 1 Real Numbers 17

NAME DATE PERIOD Lesson 8 Extra Practice Roots Find each square root. 1. 9 3 2. 81 9 3. 625 25 4. 36 6 5. 169 13 6. 144 12 7. 961 31 8. 324 18 9. 225 15 10. 4 2 11. 529 23 12. 484 22 13. 0.04 0.2 14. 2.25 1.5 15. 0.01 0.1 16. 0.09 0.3 17. 0.49 0.7 18. 1.69 1.3 19. 4 9 2 3 20. 81 64 9 8 21. 25 81 5 9 Course 3 Chapter 1 Real Numbers

NAME DATE PERIOD Lesson 9 Reteach Estimate Roots Most numbers are not perfect squares or cubes. You can estimate roots for these numbers. Example 1 Estimate 204 to the nearest integer. The largest perfect square less than 204 is 196. The smallest perfect square greater than 204 is 225. 196 < 204 < 225 Write an inequality. 14 2 < 204 < 15 2 196 = 14 2 and 225 = 15 2. 14 2 < 204 < 15 2 Find the square root of each number. 14 < 204 < 15 Simplify. So, 204 is between 14 and 15. Since 204 is closer to 196 than 225, the best whole number estimate for 204 is 14. Example 2 Estimate 3 79.3 to the nearest integer. Copyright The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. The largest perfect cube less than 79.3 is 64. The smallest perfect cube greater than 79.3 is 125. 64 < 79.3 < 125 Write an inequality. 4 3 < 79.3 < 5 3 64 = 4 3 and 125 = 5 3. 3 64 < 3 79.3 < 3 125 Find the cube root of each number. 4 < 3 79.3 < 5 Simplify. So, 3 79.3 is between 4 and 5. Since 79.3 is closer to 64 than 125, the best whole number estimate for 3 79.3 is 4. Exercises Estimate to the nearest integer. 1. 8 2. 37 3. 14 4. 3 30 5. 3 750 6. 3 200 7. 103 8. 141 9. 14.3 10. 51.2 11. 3 340.8 12. 3 7.5 Course 3 Chapter 1 Real Numbers 19

NAME DATE PERIOD Lesson 9 Extra Practice Estimate Roots Estimate to the nearest integer. 1. 229 15 2. 63 8 3. 290 17 4. 27 5 5. 333 18 6. 23 5 7. 96 10 8. 200 14 9. 117 11 10. 47 7 11. 1.3 1 12. 8.4 3 13. 18.35 4 14. 25.7 5 15. 14.1 4 16. 15.3 4 17. 32.7 6 18. 55.2 7 Course 3 Chapter 1 Real Numbers