STAT 0086 Ohio State University Algebra and Exponent Rules Questionnaire

Homework 15 – Exponent Rules1. Simplify the expressions.
a) (𝑎5 )2 =
b) (32 )−2 =
c) 𝑥 −5 ∙ 𝑥 3 =
d) (
5𝑥 2 𝑦 5
𝑥𝑦
2
) =
e) −𝑥 2 (2𝑥 5 − 𝑥 2 + 2𝑥) =
Name: _____________________________
STAT0086 Co-Requisite – Lesson 15
Exponent Rules
(𝑎𝑚 )𝑛 = 𝑎𝑚∙𝑛
Power Rule:
(𝑥 2 )5 = 𝑥 2 ∙ 𝑥 2 ∙ 𝑥 2 ∙ 𝑥 2 ∙ 𝑥 2 = 𝑥10
Example: Simplify the expressions.
a)
(𝑎8 )2
Power of a Product:
b)
(25 )2
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎
Example: Simplify the expressions.
a)
(𝑎2 𝑏)3
b)
(2𝑥𝑦 5 )2
c)
(−2𝑥𝑦)4
d)
4(−3𝑥 2 )3
Power of a Quotient:
𝑥 𝑎
𝑥𝑎
(𝑦) = 𝑦 𝑎
Example: Simplify the expressions.
𝑥
(𝑦 2)
a)
3
b)
−1 8
(𝑥)
Review of Exponent Rules
1. The exponent 1:
2. The exponent 0:
3. The product rule:
4. The quotient rule:
𝑎1 = 𝑎
𝑎0 = 1
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚 + 𝑛
𝑎𝑚
= 𝑎𝑚 − 𝑛
𝑎𝑛
−𝑛
1
5. Negative exponents:
6. Power rule:
7. Power of a product:
𝑎 = 𝑎𝑛
(𝑎𝑚 )𝑛 = 𝑎𝑚∙𝑛
(𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛
8. Power of a quotient:
(𝑏 ) = 𝑏 𝑛
𝑎 𝑛
𝑎𝑛
Example: Use one or more of the exponent rules to simplify the expressions.
a)
𝑥 −8 ∙ 𝑥 3
c)
(
𝑚2 𝑛 3
𝑚𝑛
b)
(𝑎−3 )4
d)
(𝑦 5)
2
)
2𝑥 −2
e)
−3𝑥 2 (2𝑥 3 + 5𝑥)
f)
𝑥(3𝑥 + 4) + 7(3𝑥 + 4)
g)
(3𝑥 6 𝑦 −11 )0
h)
−2𝑥 4 (𝑥 3 − 𝑥 2 + 2𝑥)
Homework 14 – Evaluating Exponents
and Exponent Rules
Name: _____________________________
1. Evaluate each expression:
a)
53
c)
(−10)5
b)
(−5)4
2. Simplify each expression:
a)
(2𝑥𝑦 3 )(𝑥 2 𝑦)
b)
c)
(−7)−3
d)
−8𝑥 5
8𝑥 4
(16𝑦 𝑜 )2𝑦 −2
STAT0086 Co-Requisite – Lesson 14
Evaluating Exponents and Exponent Rules
Exponent Form
3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = 37
Example 1: Write the following products in exponent form.
a)
2∙2∙2∙2∙2
b)
8∙8∙8∙𝑥∙𝑥∙𝑥∙𝑥
Evaluating Exponents
34 = 3 ∙ 3 ∙ 3 ∙ 3 = 81
Example 2: Evaluate the following expressions.
a)
83
b)
107
c)
42
d)
29
e)
(−3)4
f)
−34
Product Rule:
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚 + 𝑛
𝑥 4 ∙ 𝑥 5 = (𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥) ∙ (𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥)
=𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥∙𝑥
= 𝑥9
Example 3: Simplify the following expressions using the product rule.
a)
𝑥3 ∙ 𝑥5
c)
−2𝑥 3 (4𝑥)
𝑎𝑚
Quotient Rule:
𝑎𝑛
b)
𝑦 ∙ 𝑦2
= 𝑎𝑚 − 𝑛
𝑎8 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎
=
𝑎3
𝑎∙𝑎∙𝑎
=
𝑎∙𝑎∙𝑎∙𝑎∙𝑎
1
= 𝑎5
Example 4: Simplify the following expressions using the quotient rule.
a)
c)
75
74
−20𝑏 8
4𝑏 3
b)
d)
𝑥3
𝑥
𝑎12 𝑏 2
𝑎4 𝑏
1
𝑎−𝑛 = 𝑎𝑛
Negative Exponents:
Simplify
𝑥3
𝑥5
using the quotient rule.
Simplify
𝑥3
𝑥5
using cancellation of factors.
Example: Simplify the following expressions. The final answer should have only positive
exponents.
a)
𝑥 −1
b)
2−3
c)
8𝑥 −2
d)
(8𝑥)−2
e)
(−3)−5
Zero Exponent:
Simplify
𝑎8
𝑎8
𝑥0 = 1
using the quotient rule.
Simplify
𝑎8
𝑎8
using cancellation of factors.
Example: Simplify the expressions.
a)
3𝑥 0
b)
(9𝑥𝑦)0