4-3 Multiplying and Dividing Monomials
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1 Find each product. Express using positive exponents x 10 x x w 2 (5w 7 ) w 9 7. m 8 m y 4 y 12 y 8 esolutions Manual - Powered by Cognero Page 1
2 Find each quotient. Express using positive exponents r b 11 b 2 b n 5 n 4 n 9 esolutions Manual - Powered by Cognero Page 2
3 17. The Grand Canyon is approximately 2 9 kilometers long. Mariner Valley is a canyon on Mars that is approximately 2 12 kilometers long. About how many times longer is Mariner Valley than the Grand Canyon? Divide the length of Mariner Valley by the length of the Grand Canyon. 20. ( 2) 3 ( 2) 2 ( 2) a 7 a 2 Mariner Valley is about 2 3 or 8 times longer than the Grand Canyon. 2 3 or 8 times 18. A snake is 2 5 inches long. An earthworm is 2 1 inch long. About how many times as long is the length of the snake than the length of the earthworm? Divide the length of the snake by the length of the earthworm. The snake is 2 6 or 64 times longer than the earthworm. a (t 3 )(t 3 ) t or 64 times Find each product. Express using positive exponents or or 1 esolutions Manual - Powered by Cognero Page 3
4 25. c 2 c p 7 9p (w 4 )(w 6 ) 54p m 5 ( 4m 6 ) w (10x)(4x 7 ) 4m 30. ( 8s 3 )( 3s 4 ) 24s 7 Find each quotient. Express using positive exponents esolutions Manual - Powered by Cognero Page 4
5 a k 3 esolutions Manual - Powered by Cognero Page 5
6 ( n) 6 ( n) 4 y ( 1.5) 8 ( 1.5) Sound intensity is measured in decibels. The decibel scale is based on powers of ten, as shown. ( 1.5) r 20 r 6 r 14 a. How many times as intense is a rock concert as a normal conversation? b. How many times as intense is a vacuum cleaner as a person whispering? a. Divide the intensity of a rock concert by the intensity of a normal conversation. A rock concert is 10 5 or 100,000 times as intense as a normal conversation. b. Divide the intensity of a vacuum cleaner by the intensity of a person whispering. A vacuum cleaner is 10 6 or 1,000,000 times as intense as a person whispering. a or 100,000 times b or 1,000,000 times esolutions Manual - Powered by Cognero Page 6
7 44. A large beetle can be 2 7 millimeters long. One of the smallest beetles can be 2 2 millimeter long. How many times as great is the length of the large beetle than the length of the small beetle? Divide the length of the large beetle by the length of the small beetle. 46. The largest sea cucumbers are more than 10 2 times longer than the smallest sea cucumbers. If the smallest species of sea cucumbers are about 10 millimeters long, find the approximate length of the largest sea cucumbers. The length of the large beetle is 2 9 or 512 times as great as the length of the small beetle. 2 9 or 512 times 45. A person weighing 5 3 pounds can experience forces 5 times their body weight while running. Find to find the number of pounds exerted on each foot of the person while running. The approximate length of the largest sea cucumbers is 10 3 or 1000 millimeters or 1000 mm 47. A nurse draws a sample of blood. A cubic millimeter of the blood contains 22 5 red blood cells and 22 3 white blood cells. Compare the number of red blood cells to the number of white blood cells as a fraction. Explain its meaning. 5 4 or 625 pounds are exerted on a person s foot while running. 5 4 or 625 lb The fraction of the number red blood cells to white blood cells is. For every 484 red blood cells, there is one white blood cell. 48. ; Sample answer: For every 484 red blood cells, there is one white blood cell. Find each missing exponent. When multiplying powers, the exponents are added = 3, so (5 1 )(5 2 ) = 5 3. The missing exponent is 1. 1 esolutions Manual - Powered by Cognero Page 7
8 When multiplying powers, the exponents are added ( 2) = 8, so (9 10 )(9 2 ) = 9 8. The missing exponent is 2. 2 When multiplying powers, the exponents are added = 19, so (a 12 )(a 7 ) = a 19. The missing exponent is When dividing powers, the exponents are subtracted. 10 ( 3) = 13, so c 10 c 3 = c 13. The missing exponent is Multiple Representations In this problem, you will investigate area and volume. The formulas A = s 2 and V = s 3 can be used to find the area of a square and the volume of a cube, respectively, with side length s. a. Table Copy and complete the table shown When dividing powers, the exponents are subtracted = 8, so. The missing exponent is b. Words How are the area and volume each affected if the side length is doubled? tripled? c. Words How are the area and volume each affected if the side length is squared? cubed? 52. When dividing powers, the exponents are subtracted. 7 7 = 0, so. The missing exponent is 7. 7 a. Side Area of Square Volume of Cube Length (units) (units 2 ) (units 3 ) s s 2 s 3 2s 3s (2s) 2 = (2s)(2s) = 4s 2 (3s) 2 = (3s)(3s) = 9s 2 s 2 (s 2 ) 2 = (s 2 )(s 2 ) = s 4 s 3 (s 3 ) 2 = (s 3 )(s 3 ) = s 6 (2s) 3 = (2s)(2s)(2s) = b. If the side length is doubled, the area is quadrupled and the volume is multiplied by 8. If the side length is 8s 3 (3s) 3 = (3s)(3s)(3s) = 27s 3 (s 2 ) 3 = (s 2 )(s 2 )(s 2 ) = s 6 (s 3 ) 3 = (s 3 )(s 3 )(s 3 ) = s 9 esolutions Manual - Powered by Cognero Page 8
9 tripled, the area is multiplied by 3 2 or 9 and the volume is multiplied by 3 3 or 27. c. If the side length is squared, the area and volume are squared. If the side length is cubed, the area and volume are cubed. a x 3 y ( 2xy 2 ) 20x 4 y b. If the side length is doubled, the area is quadrupled and the volume is multiplied by 8. If the side length is tripled, the area is multiplied by 3 2 or 9 and the volume is multiplied by 3 3 or 27. c. If the side length is squared, the area and volume are squared. If the side length is cubed, the area and volume are cubed. Find each product or quotient. Express using exponents. 55. ab 5 8a 2 b n 6 8a 3 b 10 s 4 esolutions Manual - Powered by Cognero Page 9
10 59. 7x 2 y 5 3y 62. Find the Error Noah is multiplying (4a 2 )(4a 3 ). Find his error and correct it. 21x 2 y a 7 ( 4a 2 b 6 ) Noah did not multiply the coefficients together. Noah did not multiply the coefficients together; 16a Persevere with Problems Use the Quotient of 48a 9 b Reason Abstractly Write two algebraic expressions whose quotient is x 5. Powers Property and the equation to show that a nonzero number raised to the zero power equals 1. Sample answer: By the Quotient of Powers Product, or a 0 for a 0. Since, then a 0 = 1. So, any number raised to the zero power must equal 1. So, a sample answer is x 7 and x 2. Sample answer: x 7, x 2 Sample answer: By the Quotient of Powers Property, or a 0 for a 0. Since, then a 0 = 1. So, any nonzero number raised to the zero power must equal Use a Counterexample True or false. For any integer a, ( a) 2 = a 2. If true, explain your reasoning. If false, give a counterexample. The statement is false. If a = 3, then ( 3) 2 = 9, but 3 2 = 9. False; if a = 3, then ( 3) 2 = 9, but 3 2 = 9. esolutions Manual - Powered by Cognero Page 10
11 65. Building on the Essential Question Explain how to use division of powers to divide large numbers. Sample answer: Write the numbers as powers with the same base. Then divide by subtracting the exponents. Sample answer: Write the numbers as powers with the same base. Then divide by subtracting the exponents. 66. In the metric system, one meter is equal to 10 2 centimeters. One kilometer is 10 3 meters. How many centimeters are in one kilometer A 1000 B 10,000 C 100,000 D 1,000, Which of the following expressions is equivalent to the product of 5a 3 and 3a 8? A 8a 11 B 8a 24 C 15a 11 D 15a 24 So, the correct answer is choice C. C = = 10 5 or 100,000 So, the correct answer is choice C. C 67. Which of the following expressions is equivalent to ? F 12 7 G 12 3 H 12 3 J 12 7 So, the correct answer is choice F. F esolutions Manual - Powered by Cognero Page 11
12 69. SHORT RESPONSE The formula A = bh can be used to find the area of a triangle with base b and height h. Write an expression in simplest form to represent the area of the triangle shown below. Show your work (5) The product of two integers with different signs is negative. So, 14(5) = ( 6) The product of two integers with different signs is negative. So, 12( 6) = Which numbers in the table can be expressed as whole numbers raised to a power? Name the cities and express the numbers as powers. Chicago: 400 = 20 2 ; Evansville: 100 = 10 2 ; Nashville: 125 = 5 3 ; Paducah: 25 = 5 2 ; St. Louis: 225 = 15 2 Find each product The product of two integers with the same sign is positive. So, = 100. Chicago: 20 2 ; St. Louis: 15 2 ; Nashville: 5 3 ; Evansville: 10 2 ; Paducah: esolutions Manual - Powered by Cognero Page 12
13 74. The nameplate on an office door measures 16 inches long and 4 inches wide. What is the area of the nameplate? To find the area of the nameplate, multiply the length by the width = 64 So, the area of the nameplate is 64 square inches. 64 in The top of an ocean buoy is 24 inches above sea level. The bottom of the buoy is 6 inches below sea level. Write and evaluate an expression to find the distance from the top of the buoy to the bottom of the buoy. The distance of the top of the buoy above sea level can be represented by the integer 24. The distance of the bottom of the buoy below sea level can be represented by the integer 6. To find the distance from the top of the buoy to the bottom of the buoy, find the absolute value of the difference between 24 and w Find each sum or difference. Write in simplest form. The distance from the top of the buoy to the bottom of the buoy is 30 inches. ; 30 inches Write each expression using a positive exponent esolutions Manual - Powered by Cognero Page 13
14 Evaluate each expression if a = 3, b = 7, and c = esolutions Manual - Powered by Cognero Page 14
15 85. 7ab + c a + 10b 40 esolutions Manual - Powered by Cognero Page 15
9-3 Multiplying and Dividing Monomials
Find each product. Express using exponents. 1. 2 4 2 6 2. 8 5 8 3. x 10 x 6 4. w 2 (5w 7 ) 5. Find each quotient. Express using exponents. 6. 7 9 7 esolutions Manual - Powered by Cognero Page 1 7. 8. b
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