Mount Pearl Senior High
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Unit 2: Powers and Exponent Laws
2.1. What is a Power?
Repeated multiplication can be represented by a power. When an integer,
other than 0, can be written as the product of equal factors, we can write
the integer as a power.
In the example 23:
 23 is called a power
 The number 2 is called the base because all the factors are 2’s
 The number 3 is called the exponent because that is the number
of bases we multiply together.
Examples:
24 = 2 × 2 × 2 × 2 is written as repeated multiplication
24 = 16 is written in standard form
Note 1: Brackets are important
Ex.  2   2   2 can be written as  2 2 2
The brackets here tell us the base is −2
When there is an EVEN number of negatives the product is positive.
When there are an odd number of negatives the product is negative.
Note 2: When there are no brackets
Ex. −24 = −(2 × 2 × 2 × 2) = −16
There are no brackets therefore the base is 2. The negative applies
to the entire expression.
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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Note 3:
The base and the exponent are not interchangeable
Ex. Does 23 = 32 ?
23 = 8
32 = 9
No they do not equal each other as the base and exponent are not
interchangeable.
Note 4:
To evaluate a power is to write the answer in standard form.
Ex 1.34 = 81
Ex 2. (−3)3 = −27
An exponent of 2 results in a square number:
32 is modeled by the area of a square with a side length of 3
Area of square
=3x3
= 32
= 9 square units
An exponent of 3 results in a cube number:
2 3 is modeled by the volume of a cube with a side length of 2
Volume of cube
=2x2x2
= 23
= 8 cubic units
Discuss p. 55 #2
Set pp. 55-57 #4, 5, 6abc, 7-9, 11-14, 16, 18, 19
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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2.2. Powers of Ten and Zero Exponent
Use a base of 2 to complete the table below:
Exponent
Power
Repeated multiplication
25
24
23
22
21
5
4
3
2
1
2x2x2x2x2
2x2x2x2
2x2x2
2x2
2
Standard
form
32
16
8
4
2
Look for the patterns in the columns.
The exponent decreases by one each time.
Each time the exponent decreases, the standard form is divided by two.
This pattern suggests that 20 = ______
A power with exponent 0 is equal to 1.
Zero Exponent Law
A power with any base (other than 0) and an exponent of zero equals 1!
Ex.
20 = 1
2346254765780 = 1
(-3)0 = 1
-40 = -1
Note the use of brackets!
See example 1 p. 59
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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Additional examples:
Evaluate the following:
a)
3 + 20 = 3 + 1 = 4
b)
30 + 20 = 1 + 1 = 2
c)
(3 + 2)0 = (5)0 = 1
d)
-30 + 2 = -1 + 2 = 1
e)
-30 + (-2)0 = -1 + 1 = 0
f)
-(3 + 2)0 = -(5)0 = -1
Writing Numbers in Expanded Form Using Powers of 10
Recall Place Value:
Hundred
Millions
Ten
Millions
Millions
Hundred
Thousands
2
Ten
Thousands
Thousands
Hundreds
Tens
Ones
2
3
9
4
0
6
1
1
5
4
8
0
1
7
5
3
6
8
0
Write the digits of each number below in the table above:
a)
23647
b)
9185
c)
84103
d)
200516
Now recall that numbers can be written in expanded form as below:
23647
= (2 x 10 000) + (3 x 1000) + (6 x 100) + (4 x 10) + (7 x 1)
9185
= (9 x 1000) + (1 x 100) + (8 x 10) + (5 x 1)
84103
= (8 x 10 000) + (4 x 1000) + (1 x 100) + (0 x 10) + (3 x 1)
= (8 x 10 000) + (4 x 1000) + (1 x 100) + (3 x 1)
200516 = (2 x 100 000) + (0 x 10 000) + (0 x 1000) + (5 x 100)
+ (1 x 10) + (6 x 1)
= (2 x 100 000) + (5 x 100) + (1 x 10) + (6 x 1)
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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Also note that each place value can be written as a power of 10 as below.
Hundre
d
Millions
Ten
Million
s
Million
s
Hundred
Thousand
s
Ten
Thousand
s
Thousand
s
Hundred
s
Ten
s
One
s
108
107
106
105
104
103
102
101
100
Now we can rewrite the examples above using powers of ten:
23647
= (2 x 10 000) + (3 x 1000) + (6 x 100) + (4 x 10) + (7 x 1)
= (2 x 104) + (3 x 103) + (6 x 102) + (4 x 101) + (7 x 100)
9185
= (9 x 1000) + (1 x 100) + (8 x 10) + (5 x 1)
= (9 x 103) + (1 x 102) + (8 x 101) + (5 x 100)
84103
= (8 x 10 000) + (4 x 1000) + (1 x 100) + (0 x 10) + (3 x 1)
= (8 x 10 000) + (4 x 1000) + (1 x 100) + (3 x 1)
= (8 x 104) + (4 x 103) + (1 x 102) + (3 x 100)
200516 = (2 x 100 000) + (0 x 10 000) + (0 x 1000) + (5 x 100)
+ (1 x 10) + (6 x 1)
= (2 x 100 000) + (5 x 100) + (1 x 10) + (6 x 1)
= (2 x 105) + (5 x 102) + (1 x 101) + (6 x 100)
Extra Examples:
a)
273 = (2 x 102) + (7 x 101) + (3 x 100)
b)
3907 = (3 x 103) + (9 x 102) + (7 x 100)
Discuss p. 61 #1
Set pp. 61-62 #4-8, 9ace, 10acf, 13, 14
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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2.3. Order of Operations with Powers
When subtracting
Remember to
“Add the Opposite”
Let’s go back to the basics!!!
Adding integers
Subtracting Integers
(+5) + (+2) = (+7)
(+5) + (−2) = (+3)
(−10) + (+2) = (−8)
(−10) + (−4) = (−14)
(+5) − (+2) = (+3)
(−6) − (+2) = (−8)
(−5) − (−2) = (−3)
(+10) − (−2) = (+12)
Multiplying Integers
(+3)(+3) = +9
(+5)(−3) = −15
(−4)(+5) = −20
(−2)(−2) = +4
When multiplying or dividing:
(+)(+) = (+)
(−)(−) = (+)
same signs is
positive
Dividing Integers
(+)(−) = (−)
(−)(+) = (−)
different signs
is negative
(+3) ÷ (+3) = +1
(+12) ÷ (−4) = −3
(−25) ÷ (+5) = −5
(−30) ÷ (−6) = +5
The following Order of Operations applies any time you perform more than
one operation:
 Operations in Brackets
 Exponents
 Divide and Multiply in the order they appear left to right
 Add and Subtract in the order they appear left to right
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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Examples:
a) 33 + 1
= (3)(3)(3) + 1
= 27 + 1
= 28
(b) (5 − 3)2 − 1
= (2)2 − 1
=4−1
=2
(c) [(−2)3 × 3]2
= [(−8) × 3]2
= (−24)2
= 576
(d) (32 + 30 ) ÷ (−2)1
= (32 + 30 ) ÷ (−2)1
= (9 + 1) ÷ (−2)
= (10) ÷ (−2)
= −5
Common Errors
Explain the error in each example:
52  3
a)
*Squaring means to multiply a number by itself.
 10  3 *
This person multiplied by 2 instead.
7
b)
7  2  42  4
 9  42  4 *
 36 2  4 **
 1296  4
 1292
Order of Operations were not applied properly
in two places:
* 7+ 2 should only be done after the exponent
and multiplication.
** 9 should not have been multiplied until after
the exponent.
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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c)
1  32
3  5  6   2
0
2
12  32
3  5  1  4
1 9

84
10

2
5

*
Order of Operations were not applied properly:
*In the numerator + should have been applied
before the exponent.
Discuss p. 65 #1
Set p. 66-68 #3-8, 10, 11, 12, 14-16, 18, 19, 22
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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2.4. Exponent Laws I
Product of Powers Law
Product of Powers
22 x 23
54 x 53
32 x 32
43 x 45
23 x 23
Product as
Repeated Multiplication
(2 x 2) x (2 x 2 x 2)
(5 x 5 x 5 x 5) x (5 x 5 x 5)
(3 x 3) x (3 x 3)
(4 x 4 x 4) x (4 x 4 x 4 x 4 x 4)
(2 x 2 x 2) x (2 x 2 x 2)
Product as a
Single Power
25
57
34
48
26
Conclusion: When powers that are multiplied together have the same
base, we keep the base and add the exponents.
Quotient of Powers Law
Quotient of Powers
54 ÷ 53 = 54
53
46 ÷ 42 = 46
42
27 ÷ 23 = 27
23
35 ÷ 32 = 35
32
Quotient as
Repeated Multiplication
5x5x5x5
5x5x5
4x4x4x4x4x4
4x4
2x2x2x2x2x2x2
2x2x2
3x3x3x3x3
3x3
Quotient as a
Single Power
51
44
24
33
Conclusion: When powers that are divided have the same base, we
keep the base and subtract the exponents.
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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More about the Laws of Exponents
Example: Evaluate without a calculator:
a)
2 2  2 2  52  5  103  10 2
b) 7 9  7 7  7
 2 22  521  10 32
 79  78
 2 4  53  105
7
c)
5
 55  54   4
 1 4
1

4
7
0
 16  125  100 000
 2000  100 000
 200 000 000
Note: There are no laws for adding and subtracting powers with the same
base. We use the order of operation.
Example: Evaluate without a calculator.
4
 40  40  40   4
 1  1  1  1  4
 4  4
0
1
See examples 1-3 pp. 75-76
Discuss p. 76 #3
Set pp. 76-78 #4-11, 13, 15-19
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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2.5. Exponent Laws II
Power of a Power Law
Power of a
Power
Repeated
Multiplication
(24)3
24 x 24 x 24
(32)4
32 x 32 x 32 x 32
[(-4)3]2
(-4)3 x (-4)3
[(-5)3]5
(-5)3 x (-5)3 x (-5)3 x
(-5)3 x (-5)3
Product of Factors
(2 x 2 x 2 x 2) x
(2 x 2 x 2 x 2) x
(2 x 2 x 2 x 2)
(3 x 3) x (3 x 3) x
(3 x 3) x (3 x 3)
[(-4) x (-4) x (-4)] x
[(-4) x (-4) x (-4) ]
[(-5) x (-5) x (-5) ] x
[(-5) x (-5) x (-5) ] x
[(-5) x (-5) x (-5) ] x
[(-5) x (-5) x (-5) ] x
[(-5) x (-5) x (-5) ]
Product as
a Single
Power
212
38
(-4)6
(-5)15
Conclusion: When one power is raised to another exponent, we keep
the base and multiply the exponents.
Note:
32 x 34 = 32+4 = 36
(32)4 = 32 x 32 x 32 x 32 = 32x4 = 38
Unit 2: Powers and Exponent Laws
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Mount Pearl Senior High
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Power of Product Law
Power of a
Product
(2 x 5)3
(3 x 4)2
(4 x 2)5
(5 x 3)4
Repeated Multiplication
Product of Factors
(2 x 5) x (2 x 5) x
(2 x 5)
2x2x2
x
5x5x5
3x3x4x4
4x4x4x4x4
x
2x2x2x2x2
5x5x5x5
x
3x3x3x3
(3 x 4) x (3 x 4)
(4 x 2) x (4 x 2) x (4 x 2)
x (4 x 2) x (4 x 2)
(5 x 3) x (5 x 3) x (5 x 3)
x (5 x 3)
Product of
Powers
23 x 53
32 x 42
45 x 25
54 x 34
Conclusion: When you have a product raised to an exponent, each
factor gets raised to the exponent.
Power of Quotient Law
Power of a
Quotient
Repeated Multiplication
Quotient of Factors
5
 
6
3
5 5 5
 
6 6 6
5x5x5
6x6x6
Quotient of
Powers
53
63
2
 
 3
2
2 2

3 3
2x2
3x3
22
32
Conclusion: When you have a quotient raised to an exponent, each part
of the quotient (numerator and denominator) gets raised to the exponent.
Note:
(5 + 6)3  53 + 63
and
(5 - 6)3  53 - 63
See examples 1-3 pp. 81-83
Discuss p. 83 #1
Set pp. 84-85 #4acf, 5acf, 6-12, 14-15, 16def, 17ad, 19
Unit 2: Powers and Exponent Laws
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Summary of Laws of Exponents
We have looked at six Laws of Exponents as you saw previously. Keep
in mind that there are some limitations on these laws.
a m  a n  a mn ,
(1)
Multiplication:
(2)
Division:
(3)
Zero Exponent: a 0  1
(4)
Power of Powers:
a 
(5)
Power of Product:
a  b m  a m  bm
Power of Quotient:
an
a
   n
b
b
a m  a n  a m n
m n
 a m n
n
(6)
a  bn
 a n  bn
Note:
a and b is any integer except 0
m, n are any whole numbers
m  n for law (2)
Unit 2: Powers and Exponent Laws
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