5.1: Use Properties of Exponents

5.1: Use Properties of Exponents
1.
Objectives:
To simplify numeric
and algebraic
expressions using
the properties of
exponents
Assignment:
• P. 333-335: 1, 2, 3-21
M3, 24-36 even, 39-45,
47, 50, 52, 54-56
• Further Work with
Exponents Worksheet:
Evens
You will be able to simplify expressions
with numbers and variables using
properties of exponents
Objective 1
Warm-Up, 1
Consider the algebraic expression 3 2𝑥 + 𝑦 .
We usually think of this as 3 distributed
through the parenthesis, but it also means 3
copies of 2𝑥 + 𝑦 :
2𝑥 + 𝑦
+
2𝑥 + 𝑦
+
2𝑥 + 𝑦
Multiplication is simply repeated addition
Warm-Up, 2
Now consider the algebraic expression 2𝑥 + 𝑦 3 .
What could be done to the illustration below to
represent this new expression?
2𝑥 + 𝑦
×
2𝑥 + 𝑦
×
2𝑥 + 𝑦
An exponent is simply repeated multiplication
Exponents
Exponent
3
2
Base
= 222
Exponents
mean
repeated
multiplication
Exercise 1
1. Write 24 in expanded form
2. Write 𝑥 3 in expanded form
3. Simplify 23
4. Simplify
2𝑥 2
𝑥
2
Investigation 1
In this Investigation, we
will (re)discover some
general properties of
exponents. They
include the
Multiplication and
Division Properties,
and Power
Properties.
Investigation 1: Multiplication
Step 1: Rewrite each product in expanded
form, and then rewrite it in exponential
form with a single base.
34·32
103·106
x3·x5
a2·a4
Step 2: Compare your answers to the
original product. Is there a shortcut?
Step 3: Generalize your observations by
filling in the blank: bm·bn = b-?-
Investigation 1: Multiplication
34·32
103·106
bm·bn =
x3·x5
a2·a4
Investigation 1: Powers
Step 1: Rewrite each expression without
parentheses.
(45)2
(x3)4
(5m)n
(xy)3
Step 2: Generalize your observations by
filling in the blanks:
(bm)n = b-?(ab)n = a-?-b-?-
Investigation 1: Powers
(45)2
(x3)4
(bm)n =
(ab)n =
(5m)n
(xy)3
Investigation 1: Division
Step 1: Write the numerator and denominator in
expanded form, and then reduce to eliminate
common factors. Rewrite the factors that remain
with exponents.
59
56
33 ∙ 53
3 ∙ 52
44 𝑥 6
42 𝑥 3
Step 2: Generalize your observations by filling in the
blank:
𝑏𝑚
= 𝑏 −?−
𝑛
𝑏
Investigation 1: Division
59
56
33 ∙ 53
3 ∙ 52
𝑏𝑚
=
𝑏𝑛
44 𝑥 6
42 𝑥 3
Properties of Exponents
Multiplication
Property of
Exponents
𝑏 𝒎 ∙ 𝑏 𝒏 = 𝑏 𝒎+𝒏
Power
Property of
Exponents
𝑏 𝒎 𝒏 = 𝑏 𝒎𝒏
𝑎𝑏 𝒏 = 𝑎𝒏 𝑏𝒏
Division
Property of
Exponents
𝑏𝒎
𝒎−𝒏
=
𝑏
𝑏𝒏
Exercise 2
Practice simplifying expressions.
1. 𝑥 2 𝑥 5
3.
𝑚9
𝑚6
2.
2𝑥 2 𝑦
4. 𝑎3 𝑏 7
3
Exercise 3
Simplify 3𝑥 + 2
2
Not the Power Property
Notice that when expanding 3𝑥 + 2 2 , you
don’t get to use the Power Property of
exponents to “distribute” the exponent
through the parenthesis.
The Power Property of Exponents only
works across multiplication and division
NOT addition or subtraction!
Exercise 4
Evaluate the expression.
1. 42 3
2.
3.
−32 ∙ 5
3
4.
−8 −8
2 3
9
3
Exercise 5
Use the division property of exponents to rewrite
each expression with a single exponent. Then
expand each original expression and simplify.
Compare your answers.
32
34
𝑥3
𝑥6
74
74
𝑥5
𝑥5
Properties of Exponents
Negative
Exponents
𝑏
−𝒏
1
= 𝒏
𝑏
1
𝒏
=
𝑏
𝑏 −𝒏
Zero
Exponents
𝑏0 = 1
Exercise 6
Simplify the expression.
1. 12−4
2. 𝑤 5 𝑤 −8 𝑤 6
3.
𝑐 −2
𝑑 −4
4.
20𝑥 2 𝑦 −4 𝑧 5
4𝑥 4 𝑦𝑧 3
Always Look on the Bright Side of Life…
When you simplify an algebraic expression
involving exponents, all the exponents
must be POSITIVE.
𝑎𝑏𝑐 −𝑛
𝑎𝑏
=
𝑑
𝑑𝑐 𝑛
Negative exponents in the numerator need to
go in the denominator
Always Look on the Bright Side of Life…
When you simplify an algebraic expression
involving exponents, all the exponents
must be POSITIVE.
𝑎𝑏
𝑎𝑏𝑐 𝑛
=
−𝑛
𝑑𝑐
𝑑
Negative exponents in the denominator need
to go in the numerator
Exercise 7a
Simplify the expression.
1. 𝑥 −6 𝑥 5 𝑥 3
3.
𝑠
𝑡 −4
2.
7𝑦 2 𝑧 5 𝑦 −4 𝑧 −1
4.
𝑥 4 𝑦 −2
𝑥3𝑦6
2
3
Exercise 7b
Simplify the expression
4𝑚2
3𝑛−1
∙
−6𝑚−1 𝑛5 𝑚−2
Exercise 8
The radius of Jupiter is
about 11 times greater
than the radius of
earth. How many
times as great as
Earth’s volume is
Jupiter’s volume?
𝑉=
4 3
𝜋𝑟
3
Exercise 9
The area of a rectangle is 16𝑎3 𝑏 5 𝑐 9 units2.
Find the length of the rectangle if its width
is 2𝑎2 𝑏𝑐 3 units.
Exercise 10
Let’s say the number 𝑐 × 10𝑛 is in scientific
notation. What must be true about 𝑐?
What must be true about 𝑛?
Scientific Notation
The number 𝑐 × 10𝑛 is in scientific notation
when 1 ≤ 𝑐 < 10 and 𝑛 is an integer.
Easy to multiply, divide,
and raise to powers
using the properties of
exponents
NOT so
easy to
add and
subtract
Exercise 11
Write the answer in scientific notation.
1.
4.2 × 103 1.5 × 106
2.
7.5×108 4.5×10−4
1.5×107
5.1: Use Properties of Exponents
1.
Objectives:
To simplify numeric
and algebraic
expressions using
the properties of
exponents
Insert
your
face
here
Assignment
• P. 333-335: 1, 2,
3-21 M3, 24-36
even, 39-45, 47,
50, 52, 54-56
• Further Work with
Exponents
Worksheet: Evens
“Exponents are little like me!”