4-1 Exponents Repeated Multiplication or Division Using Bases and Exponents

4-1 Exponents

Repeated Multiplicationor Division

Using Bases and Exponents

4-1 Exponents

Vocabulary/Essential Question

EQ: How can we perform repeated multiplication and division in a shortened form?

exponential form

exponent

base

power

4-1 Exponents

If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power. Both 27 and 33 represent the same power.

7

ExponentBase

2

4-1 Exponents

.

• Simple exponent examples

4-3 Properties of Exponents

The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product.

7 • 7 • 7 • 7 = 74

(7 • 7 • 7) • 7 = 73 • 71 = 74

(7 • 7) • (7 • 7) = 72 • 72 = 74

4-1 Exponents

Identify how many times 4 is a factor.4 • 4 • 4 • 4 = 44

Write in exponential form.

Additional Example 1: Writing Exponents

A. 4 • 4 • 4 • 4

Read –(63) as “ negative 6 to the 3rd power” or “negative 6 cubed”.

Reading Math

Identify how many times –6 is a factor.

(–6) • (–6) • (–6) = (–6)3

B. (–6) • (–6) • (–6)

4-1 Exponents

Identify how many times 5 and d are used as a factor.

Additional Example 1: Writing Exponents

C. 5 • 5 • d • d • d • d

Write in exponential form.

5 • 5 • d • d • d • d = 52d4

4-1 Exponents.

• Bases with negative exponents or exponents with negative bases.

4-1 Exponents

Look for a pattern in the table to extend what you know about exponents to include negative exponents.

÷ 10 ÷ 10 ÷ 10 ÷ 10

102 101 100 10–1 10–2

10 • 10

100

10

10 1

1 110

110

= 0.1

110 • 10

1100

= 0.01

4-1 Exponents

D.

= 256

= (–4) • (–4) • (–4) • (–4) (–4)4

C. (–4)4

Simplify.

Additional Example 2: Simplifying Powers

Find the product of four –4’s.

Find the product of eight 1/2’s.

4-1 Exponents

D. 97

= –25

97 = 9 • 9 • 9 • 9 • 9 • 9 • 9

= 4,782,969

= –(5) • (5) –(5)2

C. –(5)2

Simplify.

Check It Out: Example 2

Find the product of two 5’s and then make the answer negative.

Find the product of seven 9’s.

4-1 Exponents

Additional Example 1: Using a Pattern to Simplify Negative Exponents

Simplify. Write in decimal form.

A. 10–2

10–2 = 1

10 • 10

= 1100

= 0.01

B. 10–1

= = 0.1110

= 110

Extend the pattern from the table.

Multiply. Write as a decimal.


Multiply. Write as a decimal.

4-1 Exponents

Check It Out: Example 1A

10–8

=1

100,000,000

= 1

10 • 10 • 10 • 10 • 10 • 10 • 10 • 10


= 0.00000001

Multiply.

Write as a decimal.

Simplify. Write in decimal form.

4-1 Exponents

5–3

Write the power under 1; change the sign of the exponent.

Additional Example 2A: Evaluating Negative Exponents

Simplify.

Find the product of three ’s.1 5

Simplify.

4-1 Exponents

(–10)–3 Write the power under 1; change the sign of the exponent.

Additional Example 2B: Evaluating Negative Exponents

Simplify.

Find the product of three ’s. 1 –10

Simplify.

1–10 • –10 • –10

–1000 1

= –0.001

4-1 Exponents.

• Multiplying powers with the same base.


Additional Example 1: Multiplying Powers with the Same Base

A. 66 • 63

69

66 + 3

B. n5 • n7

n12

n5 + 7

Add exponents.

Add exponents.

Multiply. Write the product as one power.


D. 244 • 244

C. 25 • 2

2 6

25 + 1

248

24 4 + 4

Think: 2 = 2 1

Additional Example 1: Multiplying Powers with the Same Base Continued


Add exponents.

Add exponents.



A. 42 • 44

46

42 + 4

B. x2 • x3

x5

x2 + 3

Add exponents.

Add exponents.



Notice what occurs when you divide powers with the same base.

5 5 555

53=

5 5 5 5 5= 5 5 = 52=

5 5 55 5 5 5 5


.

• Dividing powers with the same base.


Subtract exponents.

72

75 – 3

75

73

Additional Example 2: Dividing Powers with the Same Base

Divide. Write the quotient as one power.

A.

x10

x9B.

Subtract exponents.x10 – 9

x Think: x = x1


Subtract exponents.

97

99 – 2

99

92


Divide. Write the product as one power.

A.

B. e10

e5

Subtract exponents.e10 – 5

e5


.

• Raising a power to a power


Simplify.

Multiply exponents.

Additional Example 3: Raising a Power to a Power

A. (54)2

(54)2

54 • 2

58

B. (67)9

(67)9

67 • 9

663

Multiply exponents.


Simplify.

Multiply exponents.

Additional Example 3: Raising a Power to a Power

C. D. (172)–20

(172)–20

172 • –20

17–40

Multiply exponents.

2 3

12 –3

2 3

12 • –3


Simplify.

Multiply exponents.


A. (33)4

(33)4

33 • 4

312

B. (48)2

(48)2

48 • 2

416

Multiply exponents.


Simplify.

Multiply exponents.


C. D. (134)–10

(134)–10

134 • –10

13–40

Multiply exponents.

1 4

11 –2

1 4

11• –2


Lesson Quiz

Write the product or quotient as one power.

3.

8 9n 71. n3 n4

109

105 10 4 4.

t 2

5. 32 • 33 • 35 3 10

2. 8 • 88

t9

t7

6. (m2)19 m38

7. (9-8)9 9–72 8. (104)0 1


2. Write the product as one power.

44 43 42

A. 49

B. 414

C. 424

D. 4432

Lesson Quiz for Student Response Systems

4-1 Exponents

Additional Example 3: Using the Order of Operations

4(7) + 16

Substitute 4 for x, 2 for y, and 3 for z.

Evaluate the exponent.

Subtract inside the parentheses.

Multiply from left to right.

4(24 – 32) + 42

4(16 – 9) + 16

28 + 16

Evaluate x(yx – zy) + x for x = 4, y = 2, and z = 3.

y

x(yx – zy) + xy

Add. 44

4-1 Exponents


60 – 7(7)

Substitute 5 for x, 2 for y, and 60 for z.

Evaluate the exponent.


Multiply from left to right.

60 – 7(25 – 52)

60 – 7(32 – 25)

60 – 49

Evaluate z – 7(2x – xy) for x = 5, y = 2, and z = 60.

z – 7(2x – xy)

Subtract. 11

4-1 Exponents


Evaluate 5 – (6 – 4)–3 + (– 2)0.

Evaluate the exponents.

Additional Example 3: Using the Order of Operations

5 – (6 – 4)–3 + (–2)0

= 5 – (2)–3 + (–2)0

= 5 – + 1 1 8

= 5 7 8

Add and subtract from left to right.

4-1 Exponents


Evaluate 3 + (7 – 4)–2 + (–8)0.

Evaluate the exponents.


3 + (7 – 4)–2 + (–8)0

= 3 + (3)–2 + (–8)0

= 3 + + 1 1 9

= 4 1 9

Add.


Problem of the Day

Calculate 6 to the fourth power minus 56.

1240

Documents

4-1 Exponents Repeated Multiplication or Division Using Bases and Exponents