# 5.1 Use Properties of Exponents

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```					Warm-Up, 1
Consider the algebraic expression 3(a + 2b).
We usually think of this as 3 distributed
through the parenthesis, but it also means
3 copies of a + 2b:

a + 2b   +     a + 2b    +    a + 2b
Warm-Up, 2
Make a drawing to represent the algebraic
expression 4(2a + 3b)
Warm-Up, 3
Make a drawing to represent the algebraic
expression (2a + 3b)4
5.1: Use Properties of Exponents

Objectives:
1. To simplify numeric and algebraic
expressions using the properties of
exponents
Vocabulary
As a group, define      Base         Exponent
each of these
Notation
Give an example of
each word and
leave a bit of space
revisions.
Exponents
• Exponents mean
Exponent     repeated
multiplication

2 3     = 222
Base
Exercise 1
1.   Write 24 in expanded form.
2.   Write x3 in expanded form.
3.   Simplify (23)2
4.   Simplify 2x2 ÷ x
Investigation 1
In this Investigation, we will
(re)discover some
general properties of
exponents. They include
the Multiplication and
xy
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
original product. Is there a shortcut?
Step 3: Generalize your observations by
filling in the blank: bm·bn = b-?-
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: 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.
9         3     3      4 6
5            3 5         4 x
6               2      2 3
5            3 5         4 x
Step 2: Generalize your observations by filling in
n
the blank: b       ?
b
m
b
Properties of Exponents

Multiplication      Power         Division
Property of     Properties of   Property of
Exponents        Exponents      Exponents

n
(bm)n = bmn     b      nm
bm·bn = bm+n
(ab)n = anbn      m
b
b
Exercise 2
Practice simplifying expressions.

1. x2x5                   2. (2x2y)3

m9
3.                        4. a3b7
m6
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
Exercise 4
Evaluate the expression.

1. (42)3                   2. (−8)(−8)3

3
2
3.   (−325)3              4.    
9
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

2          3          4          5
3          x          7          x
4          6          4          5
3         x           7          x
Properties of Exponents

Negative Exponents   Zero Exponents

n     1
b         n
b
b0 = 1
1
n
b n

b
Exercise 6
Simplify the expression.

1. 12−4                    2. w5w−8w6

2
 c                            2 4 5
20x y     z
3.  4                   4.
d                             4
4 x yz   3
Always Look on the Bright Side of Life…

When you simplify an algebraic expression
involving exponents, all the exponents
must be POSITIVE.
n
abc         ab
 n
d         dc
• 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.
n
ab        abc
n    
dc          d
• Negative exponents in the denominator
need to go in the numerator
Exercise 7
Simply the expression.

6 5 3
1. x x x                 2.      7y 2 z 5      y 4 z 1   
3
 s 
2
x y    4   2
3.  4                 4.  3 6 
t                       x y 
Exercise 8
greater than the
How many times as
great as Earth’s
volume is Jupiter’s
volume?                   4 3
V  r
3
Exercise 9
3 5 9
The area of a rectangle is 16a b c units2.
Find the length of the rectangle if its width
2  3
is 2a bc units.
Exercise 10
Let’s say the number c x 10n is in scientific
notation. What must be true about c?
What must be true about n?
Scientific Notation
The number c x 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.

           
7.5 108 4.5 104   

1. 4.2 103
1.5 10 
6
2.
1.5 107   
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

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