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5.1 Rules of Exponents
Learning Objectives
1.Simplify expressions using the rules of exponents.
2.Simplify expressions involving parentheses and exponents.
3.Simplify expressions involving 0 as an exponent.
Product, Quotient, and Power Rule for Exponents
If a factor is repeated multiple times, then the product can be written in exponential formAn equivalent expression written using a rational exponent. x n. The positive integer exponent n indicates the number of times the base x is repeated as a factor.
For example,
Here the base is 5 and the exponent is 4. Exponents are sometimes indicated with the caret (^) symbol found on the keyboard: 5^4 = 5*5*5*5.
Next consider the product of 2 3 and 25,
Expanding the expression using the definition produces multiple factors of the base, which is quite cumbersome, particularly when n is large. For this reason, we will develop some useful rules to help us simplify expressions with exponents. In this example, notice that we could obtain the same result by adding the exponents.
In general, this describes the product rule for exponentsxm⋅xn=xm+n; the product of two expressions with the same base can be simplified by adding the exponents.. If m and n are positive integers, then
In other words, when multiplying two expressions with the same base, add the exponents.
Example 1: Simplify: 105⋅1018.
Solution:
Answer: 1023
In the previous example, notice that we did not multiply the base 10 times itself. When applying the product rule, add the exponents and leave the base unchanged.
Example 2: Simplify: x6⋅x12⋅x.
Solution: Recall that the variable x is assumed to have an exponent of 1: x=x1.
Answer: x19
The base could be any algebraic expression.
Example 3: Simplify: (x+y)9 (x+y)13.
Solution: Treat the expression (x+y) as the base.
Answer: (x+y)22
The commutative property of multiplication allows us to use the product rule for exponents to simplify factors of an algebraic expression.
Example 4: Simplify: 2x8y⋅3x4y7.
Solution: Multiply the coefficients and add the exponents of variable factors with the same base.
Answer: 6x12y8
Next, we will develop a rule for division by first looking at the quotient of 27 and 23.
Here we can cancel factors after applying the definition of exponents. Notice that the same result can be obtained by subtracting the exponents.
This describes the quotient rule for exponentsxmxn=xm−n; the quotient of two expressions with the same base can be simplified by subtracting the exponents.. If m and n are positive integers and x≠0, then
In other words, when you divide two expressions with the same base, subtract the exponents.
Example 5: Simplify: 12y154y7.
Solution: Divide the coefficients and subtract the exponents of the variable y.
Answer: 3y8
Example 6: Simplify: 20x10(x+5)610x9(x+5)2.
Solution:
Answer: 2x(x+5)4
Now raise 23 to the fourth power as follows:
After writing the base 23 as a factor four times, expand to obtain 12 factors of 2. We can obtain the same result by multiplying the exponents.
In general, this describes the power rule for exponents(xm)n=xmn; a power raised to a power can be simplified by multiplying the exponents.. Given positive integers m and n, then
In other words, when raising a power to a power, multiply the exponents.
Example 7: Simplify: (y6)7.
Solution:
Answer: y42
To summarize, we have developed three very useful rules of exponents that are used extensively in algebra. If given positive integers m and n, then
Product rule: xm⋅xn=xm+n
Quotient rule: xmxn=xm−n , x≠0
Power rule: (xm)n=xm⋅n
Try this! Simplify: y5⋅(y4)6.
Answer: y29
Video Solution
(click to see video)
Power Rules for Products and Quotients
Now we consider raising grouped products to a power. For example,
After expanding, we have four factors of the product xy. This is equivalent to raising each of the original factors to the fourth power. In general, this describes the power rule for a product(xy)n=xnyn; if a product is raised to a power, then apply that power to each factor in the product.. If n is a positive integer, then
Example 8: Simplify: (2ab)7.
Solution: We must apply the exponent 7 to all the factors, including the coefficient, 2.
If a coefficient is raised to a relatively small power, then present the real number equivalent, as we did in this example: 27=128.
Answer: 128a7b7
In many cases, the process of simplifying expressions involving exponents requires the use of several rules of exponents.
Example 9: Simplify: (3xy3)4.
Solution:
Answer: 81x4y12
Example 10: Simplify: (4x2y5z)3.
Solution:
Answer: 64x6y15z3
Example 11: Simplify: [5( x+y)3]3.
Solution:
Answer: 125(x+y)9
Next, consider a quotient raised to a power.
Here we obtain four factors of the quotient, which is equivalent to the numerator and the denominator both raised to the fourth power. In general, this describes the power rule for a quotient(xy)n=xnyn; if a quotient is raised to a power, then apply that power to the numerator and the denominator.. If n is a positive integer and y≠0, then
In other words, given a fraction raised to a power, we can apply that exponent to the numerator and the denominator. This rule requires that the denominator is nonzero. We will make this assumption for the remainder of the section.
Example 12: Simplify: (3ab)3.
Solution: First, apply the power rule for a quotient and then the power rule for a product.
Answer: 27a3b3
In practice, we often combine these two steps by applying the exponent to all factors in the numerator and the denominator.
Example 13: Simplify: (a b 22 c 3)5.
Solution: Apply the exponent 5 to all of the factors in the numerator and the denominator.
Answer: a5b1032c15
Example 14: Simplify: (5 x 5 ( 2x−1 ) 43 y 7)2.
Solution:
Answer: 25x10(2x−1)89y14
It is a good practice to simplify within parentheses before using the power rules; this is consistent with the order of operations.
Example 15: Simplify: (−2 x 3 y 4zx y 2)4.
Solution:
Answer: 16x8y8z4
To summarize, we have developed two new rules that are useful when grouping symbols are used in conjunction with exponents. If given a positive integer n, where y is a nonzero number, then
Power rule for a product: (xy)n=xnyn
Power rule for a quotient: (xy)n=xnyn
Try this! Simplify: (4 x 2 ( x−y ) 33y z 5)3.
Answer: 64x6(x−y)927y3z15
Video Solution
(click to see video)
Zero as an Exponent
Using the quotient rule for exponents, we can define what it means to have 0 as an exponent. Consider the following calculation:
Eight divided by 8 is clearly equal to 1, and when the quotient rule for exponents is applied, we see that a 0 exponent results. This leads us to the definition of zero as an exponentx0=1; any nonzero base raised to the 0 power is defined to be 1., where x≠0:
It is important to note that 00 is undefined. If the base is negative, then the result is still +1. In other words, any nonzero base raised to the 0 power is defined to be 1. In the following examples, assume all variables are nonzero.
Example 16: Simplify:
a. (−5)0
b. −50
Solution:
a. Any nonzero quantity raised to the 0 power is equal to 1.
b. In the example −50, the base is 5, not −5.
Answers: a. 1; b. −1
Example 17: Simplify: (5x3y0z2)2.
Solution: It is good practice to simplify within the parentheses first.
Answer: 25x6z4
Example 18: Simplify: (−8 a 10 b 55 c 12 d 14)0.
Solution:
Answer: 1
Try this! Simplify: 5x0 and (5x)0.
Answer: 5x0=5 and (5x)0=1
Video Solution
(click to see video)
Key Takeaways
•The rules of exponents allow you to simplify expressions involving exponents.
•When multiplying two quantities with the same base, add exponents: xm⋅xn=xm+n.
•When dividing two quantities with the same base, subtract exponents: xmxn=xm−n.
•When raising powers to powers, multiply exponents: (xm)n=xm⋅n.
•When a grouped quantity involving multiplication and division is raised to a power, apply that power to all of the factors in the numerator and the denominator: (xy)n=xnyn and (xy)n=xnyn.
•Any nonzero quantity raised to the 0 power is defined to be equal to 1: x0=1.
Topic Exercises
Part A: Product, Quotient, and Power Rule for Exponents
Write each expression using exponential form.
1. (2x)(2x)(2x)(2x)(2x)
2. (−3y)(−3y)(−3y)
3. −10⋅a⋅a⋅a⋅a⋅a⋅a⋅a
4. 12⋅x⋅x⋅y⋅y⋅y⋅y⋅y⋅y
5. −6⋅(x−1)(x−1)(x−1)
6. (9ab)(9ab)(9ab)(a2−b)(a2−b)
Simplify.
7. 27⋅25
8. 39⋅3
9. −24
10. (−2)4
11. −33
12. (−3)4
13. 1013⋅105⋅104
14. 108⋅107⋅10
15. 51252
16. 10710
17. 1012109
18. (73)5
19. (48)4
20. 106⋅(105)4
Simplify.
21. (−x)6
22. a5⋅(−a)2
23. x3⋅x5⋅x
24. y5⋅y4⋅y2
25. (a5)2⋅(a3)4⋅a
26. (x+1)4(y5)4⋅y2
27. (x+1)5(x+1)8
28. (2a−b)12(2a−b)9
29. (3x−1)5(3x−1)2
30. (a−5)37(a−5)13
31. xy2⋅x2y
32. 3x2y3⋅7xy5
33. −8a2b⋅2ab
34. −3ab2c3⋅9a4b5c6
35. 2a2b4c (−3abc)
36. 5a2(b3)3c3⋅(−2)2a3(b2)4
37. 2x2(x+y)5⋅3x5(x+y)4
38. −5xy6(2x−1)6⋅x5y(2x−1)3
39. x2y⋅xy3⋅x5y5
40. −2x10y⋅3x2y12⋅5xy3
41. 32x4y2z⋅3xy4z4
42. (−x2)3(x3)2(x4)3
43. a10⋅( a 6)3a3
44. 10x9( x 3)52x5
45. a6b3a2b2
46. m10n7m3n4
47. 20x5y12z310x2y10z
48. −24a16b12c36a6b11c
49. 16 x4(x+2)34x(x+2)
50. 50y2(x+y)2010y(x+y)17
Part B: Power Rules for Products and Quotients
Simplify.
51. (2x)5
52. (−3y)4
53. (−xy)3
54. (5xy)3
55. (−4abc)2
56. (72x)2
57. (−53y)3
58. (3abc)3
59. (−2xy3z)4
60. (5y(2x−1)x)3
61. (3x2)3
62. (−2x3)2
63. (xy5)7
64. (x2y10)2
65. (3x2y)3
66. (2x2y3z4)5
67. (−7ab4c2)2
68. [x5y4( x+y)4]5
69. [2y( x+1)5]3
70. (a b 3)3
71. (5 a 23b)4
72. (−2 x 33 y 2)2
73. (− x 2 y 3)3
74. (a b 23 c 3 d 2)4
75. (2 x 7y ( x−1 ) 3 z 5)6
76. (2x4)3⋅(x5)2
77. (x3y)2⋅(xy4)3
78. (−2a2b3)2⋅(2a5b)4
79. (−a2b)3(3ab4)4
80. (2x3( x+y)4)5⋅(2x4( x+y)2)3
81. (−3 x 5 y 4x y 2)3
82. (−3 x 5 y 4x y 2)2
83. (−25 x 10 y 155 x 5 y 10)3
84. (10 x 3 y 55x y 2)2
85. (−24a b 36bc)5
86. (−2 x 3y16 x 2y)2
87. (30a b 33abc)3
88. (3 s 3 t 22 s 2t)3
89. (6x y 5 ( x+y ) 63 y 2z ( x+y ) 2)5
90. (−64 a 5 b 12 c 2 ( 2ab−1 ) 1432 a 2 b 10 c 2 ( 2ab−1 ) 7)4
91. The probability of tossing a fair coin and obtaining n heads in a row is given by the formula P=(12)n. Determine the probability, as a percent, of tossing 5 heads in a row.
92. The probability of rolling a single fair six-
93. If each side of a square measures 2x3 units, then determine the area in terms of the variable x.
94. If each edge of a cube measures 5x2 units, then determine the volume in terms of the variable x.
Part C: Zero Exponents
Simplify. (Assume variables are nonzero.)
95. 70
96. (−7)0
97. −100
98. −30⋅(−7)0
99. 86753090
100. 52⋅30⋅23
101. −30⋅(−2)2⋅(−3)0
102. 5x0y2
103. (−3)2x2y0z5
104. −32(x3)2y2(z3)0
105. 2x3y0z⋅3x0y3z5
106. −3ab2c0⋅3a2(b3c2)0
107. (−8xy2)0
108. (2 x 2 y 3)0
109. 9x0y43y3
Part D: Discussion Board Topics
110. René Descartes (1637) established the usage of exponential form: a2, a3, and so on. Before this, how were exponents denoted?
111. Discuss the accomplishments accredited to Al-
112. Why is 00 undefined?
113. Explain to a beginning student why 34⋅32≠96.
Answers
1: (2x)5
3: −10a7
5: −6(x−1)3
7: 212
9: −16
11: −27
13: 1022
15: 510
17: 103
19: 432
21: x6
23: x9
25: a23
27: (x+1)13
29: (3x−1)3
31: x3y3
33: −16a3b2
35: −6a3b5c2
37: 6x7(x+y)9
39: x8y9
41: 27x5y6z5
43: a25
45: a4b
47: 2x3y2z2
49: 4x3(x+2)2
51: 32x5
53: −x3y3
55: 16a2b2c2
57: −12527y3
59: 16x4y481z4
61: 27x6
63: x7y35
65: 27x6y3
67: 49a2b8c4
69: 8y3(x+1)15
71: 625a881b4
73: −x6y9
75: 64x42y6(x−1)18z30
77: x9y14
79: −81a10b19
81: −27x12y6
83: −125x15y15
85: −1024a5b10c5
87: 1000b6c3
89: 32x5y15(x+y)20z5
91: 318%
93: A=4x6
95: 1
97: −1
99: 1
101: −4
103: 9x2z5
105: 6x3y3z6
107: 1
109: 3y
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