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8.3 Define and Use Zero and Negative Exponents
Goal  Use zero and negative exponents.
QUESTION
How can you simplify expressions with zero or negative exponents?
EXPLORE
Evaluate powers with zero and negative exponents
STEP 1 Find a pattern
Complete the tables for the powers of 2 and 3.
Exponent, n
Value of 2n
Exponent, n
4
16
4
3
3
2
2
1
1
Value of 3n
81
As you read the tables from the bottom up, you see that each time the exponent is increased by 1,
the value of the power is multiplied by the base. What can you say about the exponents and the
values of the powers as you read the table from the top down?
STEP 2 Extend the pattern
Complete the tables using the pattern you observed in Step 1.
Exponent, n
Power, 2n
Exponent, n
Power, 3n
3
8
3
27
2
2
1
1
0
0
–1
–1
–2
–2
DRAW CONCLUSIONS Use your observations to complete these exercises
1. Find 2n and 3n for n = –3, –4, and –5.
2. What appears to be the value of a 0 for any nonzero number a?
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3. Write each power in the tables above as a power with a positive exponent. For example, you
1
can write 3–1 as 31 .
DEFINITION OF ZERO AND NEGATIVE EXPONENTS
Words
Algebra
0
a to the zero power a a is 1.
a0 = ____, a ≠ 0
Example
50 = ____
a–n is the reciprocal of an.
a–n = ____ ,a ≠ 0
2–1 = ____
an is the reciprocal of a–n.
an = ____ ,a ≠ 0
25 = ____
Example 1 - Use definition of zero and negative exponents
Evaluate the expression.
a. 2–3 =
b. (-10)° =
3
c.
1
  
4
d. 0–7 =
SUMMARY OF PROPERTIES OF EXPONENTS
Let a and b be real numbers, and let m and n be integers.
am. an = a ______
______________________________ property
(am)n = a ____
______________________________ property
(ab)m = _______
______________________________ property
m
a
= a _____ , a ≠ 0
an
m
a  = ____ b ≠ 0
 
b 
______________________________ property
______________________________ property
b
Example 2
Evaluate exponential expressions
a) (-5)4 • (-5)– 4 =
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b) (5-2)–2 =
c)
1
=
4 2
d) 32 =
31
Checkpoint Evaluate the expression.
1)
1  1
 
8 
2)
1
3 2
1
3)
b66
4) (5–1)2
Example 3 - Use properties of exponents
Simplify the expression
2𝑤 −3 𝑥
(2𝑤𝑥)2
Write your answer using only positive exponents.
=
3
(2𝑤𝑥)2
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Solution
2𝑤 −3 𝑥
Checkpoint Simplify the expression.
𝟓)
𝟔𝒇𝒈−𝟒
𝟐𝒇𝟐 𝒈
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4
6) (𝟑𝒚𝒛𝟐 )−𝟐