Negative Exponents and Zero Exponents
Maybe you have learned ”the Laws of Exponents” with positive integer exponents.(→See the right side.)
In this page we will define "Negative Exponents and Zero Exponents".
Let a≠0, n be a positive integer.
a−n= …(1)
a0=1 …(2)
(1)(2)←
When the exponent is increased by 1, the value of the expression is multiplied by the base a.
On the other hand, when the exponent is decreased by 1, the value of the expression is divided by the base a.
Thus, a0=a1÷a=1,
a−1=a0÷a=,
a−2=a−1÷a=,
a−3=a−2÷a=,
…
a−n=,

(1)←Another explanation
When m<n, the following expression have negative exponent.
am÷an==am−n
Example

Thus, a−2=
Generally, a−n=

(2)←Another explanation
When m=n, the following expression have zero exponent.
an÷an==a0
Example

Thus, a0=1
 Review of Previous Lesson
•The definition of Exponents:
Exponents (or Powers) indicate how many times the base is multiplied.
an=a×a×a×… (←n times)
Example

•The Laws of Exponents:
am×an=am+n …(3)
Example
a2×a3=(a×a)×(a×a×a)=a×a×a×a×a=a5
∴a2×a3=a2+3
am÷an= =am−n …(4)
Example
a5÷a3==a2
∴a5÷a3=a5−3
(am)n=amn …(5)
Example
(a2)3=a2×a2×a2=a6
∴(a2)3=a2×3
(ab)n=anbn …(6)
Example
(ab)3=ab×ab×ab=a3b3
∴(ab)3=a3b3












Caution!
•a−n isn’t equal to a, but equal to
•a0 isn’t equal to 0, but equal to 1

Let a≠0, n, m be any integer (positive, negative or zero), by defining
a−n= , a0=1
”The Laws of Exponents” (3)(4)(5) and (6) will work.

In the following examples, you had better write your answers like the blue marked expressions. In addition, to write it without negative exponents rewrite the last expressions like the red marked ones.

(3)→Example
Calculating with negative exponents
a−2a5=a−2+5=a3
According to the definition
a−2a5=a5=a3
These are equal. So you can write your answer like the blue marked expressions.

(3)→Example
Calculating with negative exponents
a−2a−3=a−2−3=a−5
According to the definition
a−2a−3==
These are equal. In order to write your answer without negative exponents, rewrite only the last expression like the red marked one.
a−2a−3=a−2−3=a−5=

(4)→Example
Calculating with negative exponents
a−5÷a−3=a−5−(−3)=a−2
According to the definition
a−5÷a−3=÷=×a3=
These are equal. In order to write your answer without negative exponents, rewrite only the last expression like the red marked one.
a−5÷a−3=a−5−(−3)=a−2=

(5)→Example
Calculating with negative exponents
(a−2)3=a−2×3=a−6
According to the definition
(a−2)3=()3=
These are equal. In order to write your answer without negative exponents, rewrite only the last expression like the red marked one.
(a−2)3=a−2×3=a−6=

(5)→Example
Calculating with negative exponents
(a−2)−3=a−2×(−3)=a6
According to the definition
(a−2)−3=()−3===a6
These are equal. So you can write your answer like the blue marked expressions.

(6)→Example
Calculating with negative exponents
(a−2b3)−4=a−2×(−4)b3×(−4)=a8b−12
According to the definition
(a−2b3)−4=(b3)−4 = = =
These are equal. In order to write your answer without negative exponents, rewrite only the last expression like the red marked one.
(a−2b3)−4=a−2×(−4)b3×(−4)=a8b−12 =
QuestionChoose the equivalent expressions.
(1) 2−3

0 1 2 4 8

2 2 2
(2) 20

0 1 2 4 8

2 2 2
(3) ()−3

0 1 2 4 8

2 2 2
(4) a−2(a≠0)

0 1 a a2 a3

a a
(5) a0(a≠0)

0 1 a a2 a3

a a
(6) a2a−4(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(7) a2÷a−4(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(8) (a2)−4(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(9) (a−2)−4(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(10) a2÷a−2×a4(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(11) (a2)−4÷(a−3)2(a≠0)

0 1 a a2 a4 a6 a8




a a a a
(12) (3−2×2)−1÷(3×2−1)−2

0 1 8 9
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