Math 90: Unit 2 – Powers and Exponent Laws
2.1 - What is a Power?
A. Investigate:
1. Use tiles to make as many different-sized larger squares. Record
your results in a table:
Number
of tiles
Area
(sq. units)
Side Length
(units)
Area as a
Product
2. Use cubes to make as many different-sized larger cubes. Record
your results in a table:
Number
of cubes
Volume
(cubic units)
Edge Length
(units)
Volume as a
Product
What patterns do you see in the tables?
Use the patterns to predict the areas of the next 3 squares and the
volumes of the next 3 cubes.
How are these areas and volumes the same? How are they different?
B. Connect
2
4
A power with an integer base and exponent 2 is a square number.
i.e. 42
in standard form:
as repeated multiplication:
as a power:
A power with an integer base and exponent 3 is a cube number.
i.e. 23
in standard form:
as repeated multiplication:
as a power:
Examples:
1. Write as a power.
a) 5 x 5 x 5 x 5 x 5 x 5
b) 4
2. Write as repeated multiplication and in standard form.
a) 25
b) 34
c) (−4)2
d) −42
e) (−4)3
f) −(−4)2
Assignment: pages 55 – 57, # 1 – 16 (need calculator for #16)
Challenge: #20 – 25 (optional)
Discuss: how to use a calculator to evaluate exponents
Assessment Focus Question #17, Reflect – in journal
2.2 - Powers of Ten and the Zero Exponent
A. Investigate
Choose a number between 1 and 10 as the base of a power.
Use your base and the exponents below to fill in the chart:
Exponent
Power
Repeated
Multiplication
Standard
Form
5
4
3
2
1
Describe any patterns in your table.
Continue the patterns to complete the entries in the last row.
Compare your tables and patterns with another pair of
students. What do you think is the value of a power with
exponent zero?
Use your calculator to check your answer for different bases.
Now write your own rule for any power with an exponent of
zero.
B. Connect
Have students complete the following chart:
Number in Words Standard Form
Power
One billion
One hundred
million
Ten million
One million
One hundred
thousand
Ten thousand
One thousand
One hundred
Ten
One
How is this chart similar to the table from the investigate?
Does it matter what the base of the power is when we evaluate a
power with exponent zero?
Zero Exponent Law:
Any base raised to the power of 0 is equal to 1.
n0 1
(n  0)
Examples:
1. Evaluate Each Expression.
a) 70
b) −70
c) (−7)0
2. Write each number as a power of 10.
a) 1000
b) 1 000 000
c) ten thousand
3. Use powers of 10 to write each number.
a) 5476
b) 3 000 248
4. Write each number in standard form.
a) (4 𝑥 105 ) + (9 𝑥 102 ) + (3 𝑥 101 ) + (8 𝑥 100 )
b) 6 𝑥 1011
Assignment: pages 61 – 62, #1 – 11all
AFQ #12, Reflect in journal
Challenge Questions: #13 – 15
2.3 - Order of Operations with Powers
A. Investigate p. 63
B. Connect
In mathematics, why do we do operations in a certain order?
What is the order we need to follow?
Examples:
1. Evaluate.
a) 42 + 32
d) (150 + 9)3 ÷ (−2)2
b) (4 + 3)2
c) [3 ∙ (−1)3 − 2]2
e) (−2)3 − (−3)2 ÷ (12)0 ∙ 22
2.3 Assignment: Pages 65 - 68, # 1 - 3, 5, 6, 7, 8, 10, 11, 12, 15
(You may want to use a calculator for #6, 10, 12)
Challenge: #20, 24, 25 (optional)
Assessment focus #17, Reflect – in math journal
Mid-unit Review page 69, #1- 10 all
Game: Operation Target Practice (need colored dice) p. 72
2.4 - Exponent Laws – Part I
A. Investigate
Product of
Powers
Product as repeated
multiplication
Product as
a Power
Quotient of
Powers
Quotient as repeated
multiplication
Quotient as
a Power
Describe the patterns in your table.
Share your patterns with another group. How are your patterns the same? How are
they different?
Use your patterns to describe a way to multiply two powers with the same base.
Use your patterns to describe a way to divide two powers with the same base:
B. Connect
Product of Powers: When multiplying powers with the
same base, add the exponents and keep the base the same.
xm  xn  xm  n
Quotient of Powers: When dividing powers with the
same base, subtract the exponents and keep the base the
same.
xm  xn  xm  n
Examples:
1. Write each expression as a power.
a)  5   5
3
b)  3   3
6
9
2. Evaluate.
a)  2    2    2 
2
7
3. Evaluate.
2
3
0
3
a) 4  2  2  3
4
4
b)  3   3   3
11
8
b) (−10)10 ÷ [(−10)3 ∙ (−10)5 ] − (−10)3
Assignment: Pages 76-78, # 1, 3, 4 - 8, 10, 13, 15, 17
AFQ #12, Reflect – in journal
2.5 Exponent Laws – Part II
A. Investigate: Complete the table. Choose your own
power of a power for the 5th & 6th rows. Choose your own
power of a product to complete the 11th & 12th rows.
Power
(24 )3
(32 )4
[(−4)3 ]2
[(−5)3 ]5
Power
As Repeated
Multiplication
24 × 24 × 24
As a Product
of Factors
As a
Power
As Repeated
Multiplication
As a Product
of Factors
As a
Product of
Powers
(2 × 5)3 (2 × 5)(2 × 5)(2 × 5)
(3 × 4)3
(4 × 2)3
(5 × 3)3
What patterns do you see in the rows?
Use the patterns to record the rule for:
 Writing the power of a power as a single power
 Writing the power of a product as a product of two powers
B. Connect
Power of a Power:
 a m   a mn
n
Power of a Product:
 ab 
m
 a mb m
Power of a Quotient:
n
n
a a
   n
b b
Examples:
1. Write as a power.
a)
4 
2
5
b)
2
3

6
 34 


c)
3
d)
 3 
2
2. Evaluate. Use a calculator when necessary.
a)   5  2 
 21 
b) 

 3 
3
3. Simplify, then evaluate each expression.

a) 3  3
7
2

2
  62  61 
3
2
3
Assignment: Pages 83 - 85
#4, 5, 6, 7, 9, 10, 11, 12, 14, 15
AFQ # 13, Reflect
Challenge #16, 19, 20, 21
Study Guide p. 86
Unit Review:
Pages 87 - 89 # 1- 4, 6 – 14, 17 – 27