Negative Exponents and the
Laws of Exponents
Module 1 Lesson 5
Words
For any integer a, and any exponent n≠0,
𝑎
Algebra
Numbers
−𝑛
1
𝑎𝑛
=
−𝑛
𝑎
−8
3
or
=
1
𝑎𝑛
=
1
38
1
𝑎−𝑛
or
or
=𝑎
𝑛
1
𝑎−𝑛
1
3−8
=
𝑛
𝑎
8
=3
Negative
Exponent
Property
• Exercise 1
• Verify the general statement
𝑥 −𝑏
=
1
𝑥𝑏
for 𝑥 = 3 and 𝑏 = −5.
• Exercise 2
• What is the value of 3 × 10−2 ?
• Exercise 3
• What is the value of 3 × 10−5 ?
• Exercise 4
• Write the complete expanded form of the decimal 4.728 in
exponential notation.
Come to the board!
6
5
7−2
1
49
8
7
−2
−4
1
16
−2−4
1
−16
10−5 ∙ 107
100
Come to the board!
9
5𝑥 4
𝑥7
5
𝑥3
10
102 𝑎−4
𝑎4
100/𝑎
8
11
2
𝑏
∙
−2
𝑏
1
12
6∙
−3
6
1
36
We accept that for positive numbers 𝑥, 𝑦 and all integers 𝑎 and 𝑏,
𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏
𝑎
𝑏
𝑥
𝑥𝑦
𝑎
= 𝑥 𝑎𝑏
NOTES! Write these down.
= 𝑥 𝑎 𝑦𝑎
We claim:
𝑥𝑎
𝑥𝑏
= 𝑥 𝑎−𝑏
𝑥 𝑎
𝑦
=
𝑥𝑎
𝑦𝑎
for all integers 𝑎, 𝑏
for any integer 𝑎
Exercise 13
Exercise 14
2
16
19
=
195
17
=
17−3
Ponder this. . .
Reflect. . .
Think about it. . .
Meditate on the beach while doing yoga. . .
• Exercise 15
• Does
7 −1
5
=
5
7
?