Download FOM - Muchin College Prep Wiki

FOM
Name:
Teacher:
FOM, Period
CW #61: Properties of Exponents
Objective: 2020s will learn and practice the properties of exponents.
Exponential Notation

For any number 𝑥, 𝑥 1 = 𝑥, and for any positive integer 𝑛 > 1, 𝑥 𝑛 is by definition:
(𝑥 ∙ 𝑥 ⋯ 𝑥).
𝑥𝑛 = ⏟

The number 𝑥 is called 𝒙 raised to the
base of 𝑥 𝑛 .

𝑥 2 is called the square of 𝑥, and 𝑥 3 is its cube.
𝑛 times
𝑛
𝒏th
power, where 𝑛 is the exponent of 𝑥 in 𝑥 𝑛 , and 𝑥 is the
Based on the information above, determine the base and exponent of the following integers:
73
10−4
𝑥6
Base:
Base:
Base:
Base:
Exponent:
Exponent:
Exponent:
Exponent:
Laws of Exponents
RULE 1: EXPONENT PRODUCT RULE
When multiplying terms with the same base, multiply the coefficients and add the exponents.
In general, if 𝑥 is any number and 𝑚, 𝑛 are positive integers, then
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
because
(𝑥 ⋯ 𝑥) × ⏟
(𝑥 ⋯ 𝑥) = ⏟
(𝑥 ⋯ 𝑥) = 𝑥 𝑚+𝑛
𝑥𝑚 × 𝑥𝑛 = ⏟
𝑚 times
𝑛 times
𝑚+𝑛 times
RULE 2: EXPONENT QUOTIENT RULE
When dividing terms with the same base, divide the coefficients and subtract the exponents.
In general, if 𝑥 is nonzero and 𝑚, 𝑛 are positive integers, then
𝑥𝑚
= 𝑥 𝑚−𝑛
𝑥𝑛
𝑥𝑦
PRACTICE: Simplify the following using the Product and Quotient Exponent Rules:
22 + 24
𝑥2 + 𝑥3
Answer:
Answer:
RULE 3: POWER OF A POWER RULE
𝑦𝑚 ÷ 𝑦 𝑧
44 ÷ 42
Answer:
Answer:
Different Bases
For any numbers 𝑥 and 𝑦, and positive integer 𝑛,
For any number 𝑥 and any positive integers 𝑚 and 𝑛,
(𝑥 𝑚 )𝑛 = 𝑥 𝑛𝑚
because
because
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
(𝑥𝑦)𝑛 = ⏟
(𝑥𝑦) ⋯ (𝑥𝑦)
(𝑥 𝑚 )𝑛 = ⏟
(𝑥 ∙ 𝑥 ⋯ 𝑥)𝑛
𝑚 times
(𝑥 ∙ 𝑥 ⋯ 𝑥) × ⋯ × ⏟
(𝑥 ∙ 𝑥 ⋯ 𝑥)
=⏟
⏟ 𝑚 times
𝑚 times
= 𝑥 𝑛𝑚 .
𝑛 times
(𝑥 ∙ 𝑥 ⋯ 𝑥) ∙ ⏟
(𝑦 ∙ 𝑦 ⋯ 𝑦)
=⏟
𝑛 times
𝑛 𝑛
𝑛 times
=𝑥 𝑦 .
𝑛 times
RULE 4: ZERO EXPONENT RULE
Negative Exponent Rule
Any number raised to the zeroth power is equal to one.
For any nonzero number 𝑥, and for any positive
1
integer 𝑛, we define 𝑥 −𝑛 as 𝑥 𝑛 .
Example:
40 = 1 and 𝑥 0 = 1
Note that this definition of negative exponents says
1
𝑥 −1 is just the reciprocal, 𝑥, of 𝑥.
As a consequence of the definition, for a nonnegative
𝑥 and all integers 𝑏, we get
𝑥 −𝑏 =
1
𝑥𝑏
PRACTICE: Simplify the following using the various Exponent Rules:
(43 )2
𝑦0
Rule:
Rule:
Answer:
Answer:
((−7)4 )8
Rule:
Rule:
Answer:
37 × 37
Rule:
Answer:
Answer:
53
58
Rule:
(4𝑥)2
4−2
Rule:
Answer:
65
65
Rule:
Answer:
26 × 2−5
Rule:
Answer:
(𝑎9 )4
12−2
𝑥 0 × 572
Rule:
Rule:
Rule:
Answer:
Answer:
Answer:
Answer:
(𝑝𝑛 )𝑟
(𝑎𝑥 𝑛 )𝑦
Rule:
Rule:
Answer:
Answer:
(−5)4 × 3−4
Rule:
Rule:
Answer:
Answer:
(6−9 )0
Rule:
Rule:
Answer:
Answer:
𝑦9
𝑦0
1
5−8
32
33
(−2)−8
(−2)−1
Rule:
Rule:
Answer:
Answer: