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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.

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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

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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 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

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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)

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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

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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.

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c) *

Discuss p. 65 #1

Set p. 66-68 #3-8, 10, 11, 12, 14-16, 18, 19, 22

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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

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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

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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

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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

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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.

(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

References

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