CP Algebra I
Homework: ________________________________________
7.2/ 7.3 Multiplication Properties of Exponents
Objectives:
To multiply powers with the same base; To raise a power to a power
To raise a product to a power
You can use a property of exponents to multiply powers with the same base. You can write a
product of powers with the same base such as 34 ∙ 32 , using one exponent.
34 ∙ 32 = (3 ∙ 3 ∙ 3 ∙ 3) ∙ (3 ∙ 3) = 36
Notice that the sum of the exponents in the expression 34 ∙ 32 = 36 . In general, if the
exponents have the same base, you can add the exponents.
Multiplying Powers: What is each expression written using each base only once?
Example 1) 124 ∙ 123
Example 2) (−5)−2 ∙ (−5)7
Example 3) 2−4 ∙ 22
Quick Check: What is each expression written using each base only once?
1) 83 ∙ 86
2) 9−3 ∙ 92 ∙ 96
Multiplying Powers in Algebraic Expressions: When variable factors have more than one
base make sure to combine only those powers with the same base. What is the simplified form
of each expression?
Example 4) 4𝑥 5 ∙ 9𝑥 −12
Example 5) 2𝑎 ∙ 9𝑏 4 ∙ 3𝑎2
Quick Check: What is the simplified form of each expression?
3) 5𝑥 4 ∙ 𝑥 9 ∙ 3𝑥
4) −4𝑐 3 ∙ 7𝑑 2 ∙ 2𝑐 −2
5) 𝑗 2 ∙ 𝑘 −2 ∙ 12𝑗
You can use properties of exponents to simplify a power raised to a power or a product raised
to a power.
(𝑥 5 )2 = 𝑥 5 ∙ 𝑥 5 = 𝑥 5+5 = 𝑥 2∙5 = 𝑥10
Notice that (𝑥 5 )2 = 𝑥 2∙5 . Raising a power to a power is the same thing as raising the base to
the product of the exponents. In general, if you simplify a power raised to the power
multiply the two exponents and keep the same base.
Simplifying a Power Raised to a Power: Write each answer in simplified form
Example 6) (𝑛4 )7
Example 7) (𝑥 −2 )3
Quick Check: Write each answer in simplified form
6) (𝑧 4 )5
7) (𝑛5 )−2
Raising an Expression to a Power:
You can use repeated multiplication to simplify an expression like (2𝑥)3
(2𝑥)3 = (2𝑥)(2𝑥)(2𝑥) = 8𝑥 3
In general, everything within the parenthesis has to be raised to the power.
Example 8) (4𝑥𝑦 2 )3
Example 9) (2𝑥 −2 𝑦)−2
Quick Check: Simplify each expression
8) (2𝑥 3 𝑦 −2 )3
9) (5𝑥 2 𝑦)−3
Simplifying Expressions with Multiplication Properties of Exponents:
Example 10) 5𝑥 5 ∙ 2𝑥𝑦 4 ∙ 8𝑥 3
Example 11) (𝑥 −2 )2 ∙ (3𝑥𝑦 2 )4
Quick Check: Simplify
10) 𝑚4 ∙ 2𝑚−3
11) 4𝑟 −3 ∙ 2𝑟 2
12) 2𝑥 3 𝑦 −3 ∙ 2𝑥 −1 𝑦 3
13) 4𝑎3 𝑏 2 ∙ 3𝑎 −4 𝑏 −3
14) (3𝑘 4 )4 ∙ 2𝑘 −3
15) (2𝑏 4 )0