DATE: ________
AIM: _____________________________________________________
DO NOW:
1. One half of 26 is:
a) 25
b) 23
c) 16
d) 13
2. One third of 39 is:
a) 38
b) 33
c) 13
d) 19
3. If we double 26, the result is:
a) 212
b) 27
c) 46
d) 412
NEGATIVE EXPONENT RULE
Definition: For any positive number 𝑥 and for any positive integer 𝑛, we define 𝑥−𝑛 =
Note that this definition of negative exponents says 𝑥−1 is just the reciprocal,
1
𝑥
1
𝑥𝑛
.
, of 𝑥.
As a consequence of the definition, for a positive 𝑥 and all integers 𝑏, we get
𝑥−𝑏 =
1
𝑥𝑏
.
WHY? LET’S LOOK AT A MODEL.
BASE 10
BASE 2
BASE _____
SUMMARY: If we have an integer base with a negative exponent, then it’s value is a _____________________
We never leave exponents negative! We must change them to a positive exponent by ________________________________
________________________________________________________________________________________________________.
EXAMPLES:
For Exercises 1-6, write an equivalent expression, in exponential notation, to the one given and simplify as much as possible.
Exercise 1
Exercise 2
5−3 =
1
=
89
Exercise 3
Exercise 4
3 ∙ 2−4 =
Let 𝑥 be a nonzero number.
𝑥 −3 =
Exercise 5
Exercise 6
Let 𝑥 be a nonzero number.
Let 𝑥, 𝑦 be two nonzero numbers.
1
=
𝑥9
𝑥𝑦 −4 =
We accept that for positive numbers 𝑥, 𝑦 and all integers 𝑎 and 𝑏,
𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏
𝑎
(𝑥 𝑏 ) = 𝑥 𝑎𝑏
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎 .
We claim
𝑥𝑎
𝑥𝑏
𝑥 𝑎
= 𝑥 𝑎−𝑏
for all integers 𝑎, 𝑏.
𝑥𝑎
(𝑦) = 𝑦𝑎
for any integer 𝑎.
Exercise 7
Exercise 8
192
=
195
𝟏𝟕𝟏𝟔
=
𝟏𝟕−𝟑
Exercise 9
SIMPLIFY then EVALUATE: 𝟑𝟑 × 𝟑𝟐 × 𝟑𝟏 × 𝟑𝟎 × 𝟑−𝟏 × 𝟑−𝟐 =
SIMPLIFY then EVALUATE : 52 × 510 × 58 × 50 × 5−10 × 5−8 =
SIMPLIFY (𝑎 ≠ 0) : 𝑎𝑚 × 𝑎𝑛 × 𝑎𝑙 × 𝑎−𝑛 × 𝑎−𝑚 × 𝑎−𝑙 × 𝑎0 =
Equation Reference Sheet
For any numbers 𝑥, 𝑦 (𝑥 ≠ 0 in (4) and 𝑦 ≠ 0 in (5)) and any positive integers 𝑚, 𝑛, the following holds:
𝒙𝒎 ∙ 𝒙𝒏 =
(1)
(𝒙𝒎 )𝒏 =
(2)
(𝒙𝒚)𝒏 =
(3)
𝒙𝒎
=
𝒙𝒏
𝒙 𝒏
( ) =
𝒚
(4)
(5)
For any positive number 𝒙 and all integers 𝒃, the following
holds:
𝒙−𝒃 =
(9)