Exponent Properties Involving Quotients
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C ONCEPT
Concept 1. Exponent Properties Involving Quotients
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Exponent Properties Involving Quotients
Learning Objectives
• Use the quotient of powers property. • Use the power of a quotient property. • Simplify expressions involving quotient properties of exponents. Use the Quotient of Powers Property
You saw in the last section that we can use exponent rules to simplify products of numbers and variables. In this section, you will learn that there are similar rules you can use to simplify quotients. Let’s take an example of a quotient, x7 divided by x4 .
x7 x · x · x · x · x · x · x x · x · x = = = x3 x4 x · x · x · x 1 You should see that when we divide two powers of x, the number of factors of x in the solution is the difference between the factors in the numerator of the fraction and the factors in the denominator. In other words, when dividing expressions with the same base, keep the base and subtract the exponent in the denominator from the exponent in the numerator. Quotient Rule for Exponents:
xn xm
= xn−m
When we have problems with different bases, we apply the quotient rule separately for each base.
x5 y3 x · x · x · x · x y · y · y x · x y = · · = x2 y = x 3 y2 x · x · x 1 1 y · y
OR
x5 y3 = x5−3 · y3−2 = x2 y x3 y2
Example 1 Simplify each of the following expressions using the quotient rule. (a)
x10 x5
(b)
a6 a
(c)
a5 b4 a3 b2
Solution Apply the quotient rule. (a)
x10 x5
= x10−5 = x5
(b)
a6 a
= a6−1 = a5
(c)
a5 b4 a3 b2
= a5−3 · b4−2 = a2 b2
Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. 1
www.ck12.org Example 2 Simplify the following expressions, leaving all powers positive. (a)
x6 x2
(b)
a5 b6 a2 b
Solution (a) Subtract the exponent in the numerator from the exponent in the denominator and leave the x’s in the denominator.
x6 = x6−2 = x4 x2 (b) Apply the rule on each variable separately.
a5 b6 = a5−2 · b6−1 = a3 b5 a2 b The Power of a Quotient Property
When we apply a power to a quotient, we can learn another special rule. Here is an example.
x3 y2
4
=
x3 y2
3 3 3 (x · x · x) · (x · x · x) · (x · x · x) · (x · x · x) x12 x x x · 2 · 2 · 2 = = 8 y y y (y · y) · (y · y) · (y · y) · (y · y) y
Notice that the power on the outside of the parenthesis multiplies with the power of the x in the numerator and the power of the y in the denominator. This is called the power of a quotient rule. n p n·p Power Rule for Quotients yxm = yxm·p Simplifying Expressions Involving Quotient Properties of Exponents
Let’s apply the rules we just learned to a few examples. • When we have numbers with exponents and not variables with exponents, we evaluate. Example 3 Simplify the following expressions. (a)
45 42
(b)
57 53
(c)
4 2 3 52
Solution In each of the examples, we want to evaluate the numbers. (a) Use the quotient rule first. 2
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Concept 1. Exponent Properties Involving Quotients
45 = 45−2 = 43 42 Then evaluate the result.
43 = 64 OR We can evaluate each part separately and then divide.
1024 = 64 16 (b) Use the quotient rule first.
57 = 57−3 = 54 53 Then evaluate the result.
54 = 625 OR We can evaluate each part separately and then reduce.
57 78125 = = 625 53 125 It makes more sense to apply the quotient rule first for examples (a) and (b). In this way the numbers we are evaluating are smaller because they are simplified first before applying the power. (c) Use the power rule for quotients first.
34 52
2 =
38 54
Then evaluate the result.
38 6561 = 54 625 OR 3
www.ck12.org We evaluate inside the parenthesis first.
34 52
2
=
81 25
2
Then apply the power outside the parenthesis.
81 25
2 =
6561 625
When we have just one variable in the expression, then we apply the rules straightforwardly. Example 4: Simplify the following expressions: (a) (b)
x12 x5
4 5 x x
Solution: (a) Use the quotient rule.
x12 = x12−5 = x7 x5 (b) Use the power rule for quotients first.
x4 x
5 =
x20 x5
Then apply the quotient rule
x20 = x15 x5 OR Use the quotient rule inside the parenthesis first.
x4 x
5
= (x3 )5
Then apply the power rule.
(x3 )5 = x15 When we have a mix of numbers and variables, we apply the rules to each number or each variable separately. 4
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Concept 1. Exponent Properties Involving Quotients
Example 5 Simplify the following expressions. (a) (b)
6x2 y3 2xy2
2a7 b3 8a3 b
2
Solution (a) We group like terms together.
6x2 y3 6 x2 y3 = · · 2 2xy2 2 x y We reduce the numbers and apply the quotient rule on each grouping.
3xy (b) We apply the quotient rule inside the parenthesis first.
2a7 b3 8a3 b
2
=
a4 b2 4
2
Apply the power rule for quotients.
a4 b2 4
2 =
a8 b4 16
In problems that we need to apply several rules together, we must keep in mind the order of operations. Fisch Video: ExponentProperties Involving Quotients
MEDIA Click image to the left for more content.
Review Questions
Evaluate the following expressions. 1. 2. 3.
56 52 67 63 314 310
5
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2 3 2 33
Simplify the following expressions. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
a3 a2 x9 5 x 3 4 3 a b a2 b x6 y12 x 2 y5 6a3 2a2 15x5 5x 4 4 18a 15a2 25y7 x6 5 2 20y6 x4 3 x y x 4 y2 2 2 10 6a · 5b3a 4b4 (3ab)2 (4a3 b4 )3 (6a2 b)4 (2a2 bc2 )(6abc3 ) 4ab2 c
Additional Practice Opportunities (including self-check)
1. FlexMath: Findthequotient ofpowers(Four Point: Basic, Six Point: Intermediate, Eight Point: Advanced)
2. Braingenie: Simplifying expressions using the quotients of powerproperty, Simplifyexpressionsusing the p owerof a quotient property, Simplifyexpressionsusing multiple exponentproperties involving quotients
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