Negative and Zero Exponents
Lori Jordan
Kate Dirga
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Printed: February 11, 2015
AUTHORS
Lori Jordan
Kate Dirga
www.ck12.org
Chapter 1. Negative and Zero Exponents
C HAPTER
1
Negative and Zero
Exponents
Here you’ll evaluate and use negative and zero exponents.
The magnitude of an earthquake represents the exponent m in the expression 10m .
Valdivia, Chile has suffered two major earthquakes. The 1575 Valdivia earthquake had a magnitude of 8.5. The
world’s largest earthquake was the 1960 Valdivia earthquake at a magnitude of 9.5.
What was the size of the 1575 earthquake compared to the 1960 one?
Source: http://en.wikipedia.org/wiki/List_of_earthquakes
Guidance
In this concept, we will introduce negative and zero exponents. First, let’s address a zero in the exponent through an
investigation.
Investigation: Zero Exponents
1. Evaluate
56
56
56
56
by using the Quotient of Powers property.
= 56−6 = 50
2. What is a number divided by itself? Apply this to #1.
56
56
=1
3. Fill in the blanks.
am
am
= am−m = a− =−
a0 = 1
Investigation: Negative Exponents
1. Expand
32
37
=
32
37
3·3
=
3·3·3·3·3·3·3
2. Evaluate
32
37
and cancel out the common 3’s and write your answer with positive exponents.
32
37
1
35
by using the Quotient of Powers property.
= 32−7 = 3−5
3. Are the answers from #1 and #2 equal? Write them as a single statement.
1
35
= 3−5
1
am =
= am
4. Fill in the blanks.
1
am
=
a−m
and
1
a−m
a− and
1
a−m
= a−
From the two investigations above, we have learned two very important properties of exponents. First, anything to
the zero power is one. Second, negative exponents indicate placement. If an exponent is negative, it needs to be
moved from where it is to the numerator or denominator. We will investigate this property further in the Problem
Set.
1
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Example A
Simplify the following expressions. Your answer should only have positive exponents.
(a)
52
55
(b)
x7 yz12
x12 yz7
(c)
a4 b0
a8 b
Solution: Use the two properties from above. An easy way to think about where the “leftover” exponents should
go, is to look at the fraction and determine which exponent is greater. For example, in b, there are more x’s in the
denominator, so the leftover should go there.
(a)
52
55
(b)
x7 yz12
x12 yz7
(c)
a4 b0
a8 b
= 5−3 =
=
1
53
=
y1−1 z12−7
x12−7
1
125
=
y0 z5
x5
=
z5
x5
= a4−8 b0−1 = a−4 b−1 =
1
a4 b
Alternate Method: Part c
a4 b0
a8 b
=
1
a8−4 b
=
1
a4 b
Example B
Simplify the expressions. Your answer should only have positive exponents.
(a)
xy5
8y−3
(b)
27g−7 h0
18g
Solution: In these expressions, you will need to move the negative exponent to the numerator or denominator and
then change it to a positive exponent to evaluate. Also, simplify any numerical fractions.
(a)
xy5
8y−3
(b)
27g−7 h0
18g
=
xy5 y3
8
=
=
3
2g1 g7
xy5+3
8
=
=
xy8
8
3
2g1+7
=
3
2g8
Example C
Multiply the two fractions together and simplify. Your answer should only have positive exponents.
4x−2 y5 −5x6 y
·
20x8 15y−9
Solution: The easiest way to approach this problem is to multiply the two fractions together first and then simplify.
4x−2 y5 −5x6 y
20x−2+6 y5+1
x−2+6−8 y5+1+9
x−4 y15
y15
·
=
−
=
−
=
−
=
−
20x8 15y−9
300x8 y−9
15
15
15x4
Intro Problem Revisit
Set each earthquake’s magnitude up as an exponential expression and divide.
2
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Chapter 1. Negative and Zero Exponents
108.5
109.5
= 10−1
1
= 1
10
1
=
10
Therefore, the size of the 1575 earthquake was
1
10
the 1960 one.
Guided Practice
Simplify the expressions.
1.
86
89
2.
3x10 y2
21x7 y−4
3.
2a8 b−4
16a−5
3 −3 0
· 4 aa4 b7b
Answers
1.
86
89
2.
3x10 y2
21x7 y−4
3.
2a8 b−4
16a−5
= 86−9 =
=
1
83
=
1
512
x10−7 y2−(−4)
7
3 −3 0
· 4 aa4 b7b =
=
x 3 y6
7
128a8−3 b−4
16a−5+4 b7
=
8a5+1
b7+4
=
8a6
b11
Vocabulary
Zero Exponent Property
a0 = 1, a 6= 0
Negative Exponent Property
1
−m and 1 = am , a 6= 0
am = a
a−m
Explore More
Simplify the following expressions. Answers cannot have negative exponents.
1.
2.
3.
4.
5.
6.
7.
82
84
x6
x15
7−3
7−2
y−9
y10
x0 y5
xy7
a−1 b8
a5 b7
14c10 d −4
21c6 d −3
3
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8.
9.
10.
11.
8g0 h
30g−9 h2
7
5x4
· yx
10y−2 x−1 y
g9 h5 18h3
· g8
6gh12
−5 4
4a10 b7
· 9a b
12a−6 20a11 b−8
−g8 h 9g15 h9
· −h11
6g−8
12.
13. Rewrite the following exponential pattern with positive exponents: 5−4 , 5−3 , 5−2 , 5−1 , 50 , 51 , 52 , 53 , 54 .
14. Evaluate each term in the pattern from #13.
15. Fill in the blanks.
As the numbers increase, you ______________ the previous term by 5.
As the numbers decrease, you _____________ the previous term by 5.
4