1-4 Powers of Monomials. Simplify using the Laws of Exponents. 1. (4 2 ) 3 SOLUTION: To find the power of a power, multiply the exponents. Simplify.
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1 Simplify using the Laws of Exponents. 1. (4 2 ) 3 2. (5 3 ) 3 3. (d 7 ) 6 4. (h 4 ) 9 5. [(3 2 ) 2 ] 2 esolutions Manual - Powered by Cognero Page 1
2 6. [(5 2 ) 2 ] 2 7. (5j 6 ) 4 8. (11c 4 ) 3 9. (6a 2 b 6 ) (2m 5 n 11 ) 6 esolutions Manual - Powered by Cognero Page 2
3 11. ( 3w 3 z 8 ) ( 5r 4 s 12 ) A shipping box is in the shape of a cube. Each side measures 3c 6 d 2 inches. Express the volume of the cube as a monomial. The formula for the volume of a cube is V = s 3, where x is the length of each side. Write the expression for the volume using the side length 3c 6 d 2. Then simplify. The volume of the box is 27c 18 d 6 cubic units. esolutions Manual - Powered by Cognero Page 3
4 14. Tamara is decorating her patio with a planter in the shape of a cube like the one shown. Find the volume of the planter. The formula for the volume of a cube is V = s 3, where x is the length of each side. Write the expression for the volume using the side length 3w 4. Then simplify. The volume of the box is 27w 12 cubic units. Copy and Solve Simplify. Show your work on a separate sheet of paper esolutions Manual - Powered by Cognero Page 4
5 17. ( 2v 7 ) 3 ( 4v 2 ) 4 To find the power of a power, multiply the exponents. Group the numbers and the variables. Simplify. 18. Identify Structure Draw a line connecting the Law(s) of Exponents you would use to simplify each of the expressions. Then simplify each one. Examine each of the expressions. (a 9 ) 3 A power is raised to a power, use the Power of a Power law. (m 8 ) ( m 4 ) The expression involves division and exponents, so use the Quotient of Powers law 5x 2 ( 7x 4 ) The expression involves multiplication and exponents, so use the Product of Powers law. 5x 2 ( 7x 4 ) = 5 ( 7) x or 35x 6 The numerator involves a product raised to a power, so use the Power of a Product law. The expression involves division and exponents, so use the Quotient of Powers law. esolutions Manual - Powered by Cognero Page 5
6 (n 6 ) 8 A power is raised to a power, use the Power of a Power law. (n 6 ) 8 = n 6 8 or n 48 esolutions Manual - Powered by Cognero Page 6
7 19. Reason Inductively The table gives the area and volume of a square and cube, respectively, with side lengths shown. a. Complete the table. b. Describe how the area and volume are each affected if the side length is doubled. Then describe how they are each affected if the side length is tripled. a. 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 9 and the volume is multiplied by 27. Persevere with Problems Solve the equation for x. 20. (7 x ) 3 = 7 15 Notice that the bases are the same. To solve for x in the exponent, find the value that makes the equation in the exponents equal. esolutions Manual - Powered by Cognero Page 7
8 21. ( 2m 3 n 4 ) x = 8m 9 n 12 Simplify the first part of the equation. Look at each of the terms on the left side of the equation and find the value of x that will make a true statement with the corresponding term on the other side of the equation. ( 2) x = 8 This is true if x = 3. m 3x = m 9 This is true if x = 3. n 4x = n 12 This is true if x = 3. When x = 3, the terms are equal. 22. Reason Inductively Compare how you would correctly simplify the expressions and. Sample answer: To simplify a., multiply 2 by 4. Then add the exponents 3 and 6 and write this sum as the final exponent on To simplify a., evaluate 2 6. Then multiply the exponents 3 and 6 and write this product as the final exponent on Simplify using the Laws of Exponents. 23. (2 2 ) 7 esolutions Manual - Powered by Cognero Page 8
9 24. (8v 9 ) (3 4 ) (m 8 ) (z 11 ) [(4 3 ) 2 ] 2 esolutions Manual - Powered by Cognero Page 9
10 29. [(2 3 ) 3 ] (14y) 4 Express the area of each square as a monomial. 31. Recall that the area of a square equals the measure of one side squared, or A = s 2. Use Power of a Power to simplify the expression. The area of the square is 64g 6 h 2 square units. esolutions Manual - Powered by Cognero Page 10
11 32. Recall that the area of a square equals the measure of one side squared, or A = s 2. Use Power of a Power to simplify the expression. The area of the square is 144d 12 e 14 square units. Express the volume of each cube as a monomial. 33. Recall that the volume of a cube equals the measure of one side cubed, or V = s 3. Use Power of a Power to simplify the expression. The volume of the cube is 125r 4 s 9 cubic units. esolutions Manual - Powered by Cognero Page 11
12 34. Recall that the volume of a cube equals the measure of one side cubed, or V = s 3. Use Power of a Power to simplify the expression. The volume of the cube is 343m 18 n 27 cubic units. Simplify. 35. (0.5k 5 ) (0.3p 7 ) 3 esolutions Manual - Powered by Cognero Page 12
13 Persevere with Problems A ball is dropped from the top of a building. The expression 4.9x 2 gives the distance in meters the ball has fallen after x seconds. Write and simplify an expression that gives the distance in meters the ball has fallen after x 2 seconds. Then write and simplify an expression that gives the distance the ball has fallen after x 3 seconds. Replace x with x 2. Simplify. Replace x with x 3. Simplify. esolutions Manual - Powered by Cognero Page 13
14 39. Manny has four pieces of carpet in the shape of a square like the one shown. He wants to use them together to carpet a portion of his basement. What is the area of the space he can cover with the carpet? Recall that the area of a square equals the measure of one side squared, or A = s 2. Use the Power of a Product law to simplify the expression. The area of the one of the squares is 4x 4 square units. Manny has four squares of carpet. 4 4x 4 = 16x Select the correct expression to represent the volume of each cube. esolutions Manual - Powered by Cognero Page 14
15 If c = the side length of a cube, then the volume can be found using V = c 3. So, the volume of the first cube can be written: By applying the same volume formula, the volume of the second cube can be represented: esolutions Manual - Powered by Cognero Page 15
16 The last cubes volume: Simplify using the Laws of Exponents The common base is 6. Add the exponents The common base is 18. Add the exponents. 43. ( 3x 11 )( 6x 3 ) Use the Commutative and Associative Properties to group the numbers and variables. The common base is x. Add the exponents. esolutions Manual - Powered by Cognero Page 16
17 44. ( 9a 4 )(2a 7 ) Use the Commutative and Associative Properties to group the numbers and variables. The common base is a. Add the exponents. 45. The table shows the heights of some United States waterfalls. What is the height of each waterfall? Simplify each expression. Bridalveil: The height of Bridalveil Falls is 620 feet. Fall Creek: The height of Fall Creek Falls is 256 feet. Shoshone: The height of Shoshone Falls is 212 feet. esolutions Manual - Powered by Cognero Page 17
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