1-8 Roots. Find each square root. SOLUTION: Find the positive square root of 16. Since 4 2 = 16, = 4.
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1 1. Find each square root. Find the positive square root of 16. Since 4 2 = 16, = Find the negative square root of 484. Since 22 2 = 484,. There is no real solution because no number times itself is equal to Find both square roots of. Since, Find the negative square root of Since = 2.56,. There is no real solution because no number times itself is equal to esolutions Manual - Powered by Cognero Page 1
2 Solve each equation. Check your solution(s). 7. v 2 = 81 Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to = 81 and ( 9)( 9) = 81 The equation has two solutions, 9 and w 2 = Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to. and The equation has two solutions, and = c 2 Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to = and ( 0.13)( 0.13) = The equation has two solutions, 0.13 and esolutions Manual - Powered by Cognero Page 2
3 Find each cube root. 10. Since 12 3 = or 1,728,. 11. Since ( 0.5) 3 = ( 0.5)( 0.5)( 0.5) or 0.125,. 12. Since,. 13. A group of 169 students needs to be seated in a square formation for a yearbook photo. Solve the equation 169 = s 2 to find how many students should be in each row. Solve the equation using the definition of square root. There cannot be a negative number of students, so there should be 13 students in each row. 14. Chloe wants to build a storage container in the shape of a cube to hold cubic meters of hay for her horse. Solve the equation = s 3 to find the length of one side of the container. Solve the equation using the definition of cube root. The length of one side of the container is 2.5 meters. esolutions Manual - Powered by Cognero Page 3
4 Persevere with Problems Given the area of the square, find the perimeter. 15. The area A of a square is A = s 2, where s represents the length of a side. Solve the equation s 2 = 121 to find the length of each side. Since the distance cannot be negative, the length of each side is 11 inches. The perimeter P of a square is P = 4s, where s represents the length of a side. Replace s with 11. P = 4s P = 4(11) P = 44 The perimeter of a square with an area of 121 square inches is 44 inches. 16. The area A of a square is A = s 2, where s represents the length of a side. Solve the equation s 2 = 25 to find the length of each side. Since the distance cannot be negative, the length of each side is 5 feet. Perimeter P of a square is P = 4s, where s represents the length of a side. Replace s with 5. P = 4s P = 4(5) P = 20 The perimeter of a square with an area of 25 square feet is 20 feet. esolutions Manual - Powered by Cognero Page 4
5 17. The area A of a square is A = s 2, where s represents the length of a side. Solve the equation s 2 = 36 to find the length of each side. 18. Since the distance cannot be negative, the length of each side is 6 meters. The perimeter P of a square is P = 4s, where s represents the length of a side. Replace s with 6. P = 4s P = 4(6) P = 24 The perimeter of a square with an area of 36 square meters is 24 meters. Persevere with Problems Find each value. Simplify using the order of operations. 19. Simplify using the order of operations. esolutions Manual - Powered by Cognero Page 5
6 20. Simplify using the order of operations. Use a calculator to find the square root of Look for a pattern by replacing x with a positive rational number. If you square the square root of a number, the result is that number. So,. 22. Reason Abstractly Based on your solutions to Exercises 18 21, write a rule that could be used to simplify the square of any square root of a number. Sample answer: In each problem the only change from the answer to the problem is that the square root and the square are gone. So, the square of any square root of a number is the same as the original number. 23. Reason Inductively Explain why is not a real number, but is. Sample answer: There are no two equal numbers that have a product of 4, because a negative number times a negative number is positive and a positive number times a positive number is always a positive. Since ( 2)( 2)( 2) = 8 it is possible to find a solution. esolutions Manual - Powered by Cognero Page 6
7 24. Reason Inductively Describe the difference between an exact value and an approximation when finding square roots of numbers that are notperfect squares. Give an example of each. 25. Sample answer: The exact value of a square root is given using the square root symbol, such as approximation is a decimal value, such as longer exact. Find the square root. Find the negative square root of 81. Since 9 2 = 81,.. An 3.6. In addition, the decimal value has be round and therefore is no 26. Find the negative square root of. Since,. 27. Find the negative square root of. Since, Find both square roots of Since = 1.44,. Find the cube root. Since ( 6) 3 = ( 6)( 6)( 6) or 216,. esolutions Manual - Powered by Cognero Page 7
8 Since ( 8) 3 = ( 8)( 8)( 8) or 512,. Since ( 10) 3 = ( 10)( 10)( 10) or 1,000,. Since ( 7) 3 = ( 7)( 7)( 7) or 343,. Solve the equation. Check your solution(s). 33. b 2 = 100 Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to = 100 and ( 10)( 10) = 100 The equation has two solutions, 10 and = c 2 Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to. and The equation has two solutions, and. esolutions Manual - Powered by Cognero Page 8
9 35. a 2 = 1.21 Solve the equation using the definition of square root. Check the solution by verifying that each value squared is equal to = 1.21 and ( 1.1)( 1.1) = 1.21 The equation has two solutions, 1.1 and Solve the equation using the definition of cube root. Check the solution by verifying that the value cubed is equal to = c 3 Solve the equation using the definition of cube root. Check the solution by verifying that the value cubed is equal to (1.1) 3 = (1.1)(1.1)(1.1) or m 3 = 8,000 Solve the equation using the definition of cube root. Check the solution by verifying that the value cubed is equal to 8,000. (20) 3 = (20)(20)(20) or 8,000 esolutions Manual - Powered by Cognero Page 9
10 39. = 5 Square both sides of the equation. Squaring a number and finding a square root are inverse operations. So,. Check the solution by replacing x with = 20 Square both sides of the equation. Squaring a number and finding a square root are inverse operations. So,. Check the solution by replacing y with = 10.5 Square both sides of the equation. Squaring a number and finding a square root are inverse operations. So,. Check the solution by replacing z with esolutions Manual - Powered by Cognero Page 10
11 42. Persevere with Problems A concert crew needs to set up some chairs on the floor level. The chairs are to be placed in a square pattern consisting of four square sections. If one of the square sections holds 900 chairs, how many chairs will there be along each length of the larger square? Draw a diagram to represent the situation. The area A of a square is A = s 2, where s represents the length of each side. Solve the equation s 2 = 900 to find the number of chairs along each side of the four square sections. The number of chairs cannot be negative, so the number of chairs along each side of the four square sections is 30 chairs. So, there will be or 60 chairs along each length of the larger square. 43. Mr. Freeman has a square cornfield. Which of the following could be the area of the cornfield if the sides are measured in whole numbers? Select all that apply. 164,000 ft 2 156,816 ft 2 174,724 ft 2 215,908 ft 2 The area A of a square is A = s 2, where s represents the length of each side. Solve the equation A = s 2 for each area. Area = 164,000 square feet To find, find two equal factors of 164,000. Start by breaking 164,000 into prime factors. Then regroup into two equal factors. 164,000 = Because there are an odd number of each factor, you cannot create two equal factors. 164,000 is not a perfect square, so is not a whole number. esolutions Manual - Powered by Cognero Page 11
12 Area = 156,816 square feet To find, find two equal factors of 156,816. Start by breaking 156,816 into prime factors. Then regroup into two equal factors. 156,816 = = ( ) ( ) So, s = or 396. This is a whole number that could represent the side length of Mr. Freeman s square cornfield. Area = 174,724 square feet To find, find two equal factors of 174,724. Start by breaking 174,724 into prime factors. Then regroup into two equal factors. 174,724= = ( )( ) So, s = or 418. This is a whole number that could represent the side length of Mr. Freeman s square cornfield. Area = 215,908 square feet To find, find two equal factors of 215,908. Start by breaking 215,908 into prime factors. Then regroup into two equal factors. 215,908 = Because there are an odd number of certain factors, you cannot create two equal factors. 215,908 is not a perfect square, so is not a whole number. esolutions Manual - Powered by Cognero Page 12
13 44. The area of each square in the figures below is 81 square units. Select the perimeter of each figure. Do any of the figures have the same perimeter? If so, explain why. The area A of a square is A = s 2, where s represents the length of a side. Solve the equation s 2 = 81 to find the length of each side. Since the distance cannot be negative, the length of each side is 9 units. The perimeter is the distance around the figure. Count the sides. The first figure has 12 sides, so the perimeter is 12s or 12(9) or 108 units. The middle figure has 10 sides, so the perimeter is 10s or 10(9) or 90 units. The last figure has 12 sides, so the perimeter is 12s or 12(9) or 108 units. Sample Answer: Figures 1 and 3 both have the same perimeter, because distance around both figures is the same. Evaluate each expression Write the power as a product. Then multiply = = 2, Write the power as a product. Then multiply = = 625 esolutions Manual - Powered by Cognero Page 13
14 Write the power as a product. Then multiply = = 3,375 Write the power as a product. Then multiply = = 1,156 Find the positive square root of 121. Then multiply. 50. Find the positive square root of 36. Then multiply. 51. Find the cube root of 8. Then multiply. 52. Find the positive square root of 144. Then multiply. esolutions Manual - Powered by Cognero Page 14
15 Express the volume of the cube as a monomial. 53. The volume V of a cube is V = s 3, where s represents the length of a side. Simplify the expression V = (4r 3 s) 3 using the Laws of Exponents. V = (4r 3 s) 3 V = 4 3 r 3( 3) s 3 V = 64r 9 s 3 So, the volume of the cube is 64r 9 s 3 cubic units. 54. The volume V of a cube is V = s 3, where s represents the length of a side. Simplify the expression V = (9m 2 n 4 ) 3 using the Laws of Exponents. V = (9m 2 n 4 ) 3 V = 9 3 m 2( 3) n 4(3) V = 729m 6 n 12 So, the volume of the cube is 729m 6 n 12 cubic units. esolutions Manual - Powered by Cognero Page 15
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