Unit 2: Powers and Exponent Laws
2.1. What is a Power?
Just as repeated addition can be represented as multiplication, 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 53:
53 is called a power
The number 5 is called the base because all the factors are 5’s
The number 3 is called the exponent because that is the number of bases we multiply together.
Examples:
Power Base Exponent Repeated
Multiplication
Standar d Form
5 2 5 x 5 25
3 5 3 x 3 x 3x 3 x 3 243
2 4 2 x 2 x 2 x 2 16
4 1 4 4
-2 3 (-2) x (-2) x (-2) -8 -3 4 (-3) x (-3) x (-3) x (-3) 81
Note 1: In the last two examples, the multiplication signs are not necessary.
Eg. can be written as
Note 2:
Notice the use of brackets in the last few examples. This indicates that the negative is part of the base and therefore part of the repeated
multiplication.
Note 3:
The base and the exponent are not interchangeable
An exponent of 2 results in a square number:
is modeled by the area of a square with a side length of 3
Area of square = 3 x 3
= 32
= 9 square units
An exponent of 3 results in a cube number:
is modeled by the volume of a cube with a side length of 2
Volume of cube = 2 x 2 x 2 = 23
= 8 cubic units
Discuss p. 55 #2
2.2. Powers of Ten and Zero Exponent
Use a base of 2 to complete the table below:
Exponent Power Repeated multiplication Standard form
5 25 2 x 2 x 2 x 2 x 2 32
4 24 2 x 2 x 2 x 2 16
3 23 2 x 2 x 2 8
2 22 2 x 2 4
1 21 2 2
What do you think 20 will equal? See Connect p. 59
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
Additional examples:
Evaluate the following:
a) 3 + 20 = 3 + 1 = 4 b) 30 + 20 = 1 + 1 = 2 c) (3 + 2)0 = (5)0 = 1
Writing Numbers in Expanded Form Using Powers of 10
Recall Place Value:
Hundred
Millions MillionsTen Millions ThousandsHundred ThousandsTen Thousands Hundreds Tens Ones
2 3 6 4 7
9 1 8 5
8 4 1 0 3
2 0 0 5 1 6
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)
Discuss p. 61 #2
Also note that each place value can be written as a power of 10 as below.
Hundred Millions
Ten Millions
Millions Hundred Thousands
Ten Thousands
Thousands Hundreds Tens Ones
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)
Also see example 2 p. 60
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
2.3. Order of Operations with Powers
Circle the correct answer to this was a skill-testing question from a competition:
5 10 15 25
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
See examples 1 & 2 pp. 64-65
Common Errors
Explain the error in each example: a)
*
b)
* **
*Squaring means to multiply a number by itself. This person multiplied by 2 instead.
Order of Operations were not applied properly in two places:
* 7+ 2 should only be done after the exponent and multiplication.
c) *
Discuss p. 65 #1
Set p. 66-68 #3-8, 10, 11, 12, 14-16, 18, 19, 22
2.4. Exponent Laws I
Product of Powers Law
Product of Powers
Product as
Repeated Multiplication
Product as a Single Power
22 x 23 (2 x 2) x (2 x 2 x 2) 25
54 x 53 (5 x 5 x 5 x 5) x (5 x 5 x 5) 57
32 x 32 (3 x 3) x (3 x 3) 34
43 x 45 (4 x 4 x 4) x (4 x 4 x 4 x 4 x 4) 48
23 x 23 (2 x 2 x 2) x (2 x 2 x 2) 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
Quotient as
Repeated Multiplication
Quotient as a Single Power 54 ÷ 53 = 5 4
53
5 x 5 x 5 x 5 5 x 5 x 5
51
46 ÷ 42 = 4 6 42
4 x 4 x 4 x 4 x 4 x 4 4 x 4
44
27 ÷ 23 = 2 7 23
2 x 2 x 2 x 2 x 2 x 2 x 2 2 x 2 x 2
24
35 ÷ 32 = 3 5 32
3 x 3 x 3 x 3 x 3 3 x 3
33
More about the Laws of Exponents
Example: Evaluate without a calculator:
a) b) c)
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.
See examples 1-3 pp. 75-76 Discuss p. 76 #3
2.5. Exponent Laws II
Power of a Power Law
Power of a Power
Repeated
Multiplication Product of Factors
Product as a Single
Power (24)3 24 x 24 x 24 (2 x 2 x 2 x 2) x
(2 x 2 x 2 x 2) x (2 x 2 x 2 x 2)
212
(32)4 32 x 32 x 32 x 32 (3 x 3) x (3 x 3) x
(3 x 3) x (3 x 3)
38
[(-4)3]2 (-4)3 x (-4)3 [(-4) x (-4) x (-4)] x
[(-4) x (-4) x (-4) ]
(-4)6
[(-5)3]5 (-5)
3 x (-5)3 x (-5)3 x
(-5)3 x (-5)3 [(-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) ]
(-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
Power of Product Law
Power of a Product
Repeated Multiplication Product of Factors Product of Powers (2 x 5)3 (2 x 5) x (2 x 5) x
(2 x 5)
2 x 2 x 2 x 5 x 5 x 5
23 x 53
(3 x 4)2 (3 x 4) x (3 x 4) 3 x 3 x 4 x 4 32 x 42
(4 x 2)5 (4 x 2) x (4 x 2) x (4 x 2) x (4 x 2) x (4 x 2)
4 x 4 x 4 x 4 x 4 x
2 x 2 x 2 x 2 x 2
45 x 25
(5 x 3)4 (5 x 3) x (5 x 3) x (5 x 3) x (5 x 3)
5 x 5 x 5 x 5 x
3 x 3 x 3 x 3
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 Quotient of Powers 5 x 5 x 5
6 x 6 x 6
5 3
63
2 x 2 3 x 3
2 2
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
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.
(1) Multiplication: ,
(2) Division:
(3) Zero Exponent:
(4) Power of Powers:
(5) Power of Product:
(6) Power of Quotient:
Note:
and is any integer except 0 are any whole numbers