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Working with Powers

• Integer Exponents• Exponent Rules• Order of Operations

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• 2 × 2 =

• 2 × 2 × 2 =

• 2 × 2 × 2 × 2 =

• How can we write these in

shorter notation?

22

32

42

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• Multiplication is a shortcut for repeated addition.

• Exponents are a shortcut for repeated multiplication.

3 + 3 + 3 + 3 + 3 = 5 × 3

3 × 3 × 3 × 3 × 3 = 53

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Write these numbers using exponential notation:

• 10,000 = 10?

• 27 = 3?

• 32 = 2?

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Integer Exponents

• Base 4• Exponent 3

• 43 is called a power

• 43 = 4 x 4 x 4 • = 64

34

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You try these . . .

• 72

• 54

• 210

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Operations with Exponents

• Multiply (x3)∙(x4)= (x ∙ x ∙ x) ∙ (x ∙ x ∙ x ∙ x) = (x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x) = x7

A5 ∙ A4 =

• Divide: xxxxxxx

3

4

xx

1x

x

3

4

mm

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Exponent Rules

aaa1an ....#1

n times

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Zero as an exponent

• 40

• 90

• 170

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Exponent Rules

1a 0#2

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a0 = 1

• M0

• (pq)0

• (2x2y)0

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Negative Exponents

• 3-1

• 7-1

• 5-2

• 2-4

• (1/2) -1

• (3/4) -2

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Exponent Rules

nn

n aa

a 11#3

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• M-2

• x-5

• (1/y)-3

nn

aa 1

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Combining Exponents

5

32

x

xxxxxxx

aman=?

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Exponent Rules

nmnm aaa #4

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• y-2y7

• m6m-6

• 2z-3z5w-6w-2

• x5x4

• b3b-6

• (2x3)(4x-2)

nmnm aaa

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Combining Exponents

3

2

5

xxxxxxxx

xx

am/an=?

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Exponent Rules

nmn

m

aaa

#5

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nmn

m

aaa

2

6

xx

7

2

yy

4

4

155mm

3

5

510

pp

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Combining Exponents

6

222

32

x

xxxxxxxxx

x

(am)n=?

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Exponent Rules

nmnm aa )(#6

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• (w4)5

• (p-3)4

• (5x4)-2

• (-3y-7)3

nmnm aa )(

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Combining Exponents3

zxy

3

33

zyx

zzzyyyxxx

zxy

zxy

zxy

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Exponent Rules

n

nnn

cba

cab

#7 (distributive rule for exponents)

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n

nnn

cba

cab

432 yx

3

5

2

nm 2

3

465

cba

2475nm

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Exponent Rules

• 1.

• 2.

• 3.

aaaan ....

10 a

nn

aa 1

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Exponent Rules

• 4.

• 5.

• 6.

• 7.

nmnm aaa

nmn

m

aaa

nmnm aa )(

n

nnn

cba

cab