Evaluating Algebraic Expressions

4-1 Exponents

Warm Up

California StandardsCalifornia Standards

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Evaluating Algebraic Expressions

4-1 Exponents

Warm UpFind the product.

6251. 5 • 5 • 5 • 5

2. 3 • 3 • 3

3. (–7) • (–7) • (–7)

4. 9 • 9

27

–343

81

Evaluating Algebraic Expressions

4-1 Exponents

AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.Also covered: AF1.2

California Standards

Evaluating Algebraic Expressions

4-1 Exponents

Vocabulary

exponential form

exponent

base

power

Evaluating Algebraic Expressions

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. 27 and 33 are equivalent.

7

ExponentBase

2

Evaluating Algebraic Expressions

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 (–6)3 as “–6 to the 3rd power" or "–6 cubed”.

Reading Math

Identify how many times –6 is a factor.

(–6) • (–6) • (–6) = (–6)3

B. (–6) • (–6) • (–6)

Evaluating Algebraic Expressions

4-1 Exponents

Identify how many times 5 and d are each 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

Evaluating Algebraic Expressions

4-1 Exponents

Identify how many times x is a factor.x • x • x • x • x = x5

Write in exponential form.

Check It Out! Example 1

A. x • x • x • x • x

Identify how many times d is a factor.

d • d • d = d3

B. d • d • d

Evaluating Algebraic Expressions

4-1 Exponents

Identify how many times 7 and b are each used as a factor.

7 • 7 • b • b = 72b2

Check It Out! Example 1

C. 7 • 7 • b • b

Write in exponential form.

Evaluating Algebraic Expressions

4-1 Exponents

A. 35

= 243

35 = 3 • 3 • 3 • 3 • 3 Find the product.

Find the product.

B.

Simplify.

Additional Example 2: Simplifying Powers

= 1

27

Evaluating Algebraic Expressions

4-1 Exponents

D. –28

= 256

–28 = –(2 • 2 • 2 • 2 • 2 • 2 • 2 • 2)

= –256

= (–4) • (–4) • (–4) • (–4) (–4)4

C. (–4)4

Simplify.

Additional Example 2: Simplifying Powers

Find the product.

Find the product. Then make the answer negative.

Evaluating Algebraic Expressions

4-1 Exponents

The expression (–4)4 is not the same as the expression –44. Think of –44 as –1 ● 44. By the order of operations, you must evaluate the exponent before multiplying by –1.

Caution!

Evaluating Algebraic Expressions

4-1 Exponents

A. 74

= 240174 = 7 • 7 • 7 • 7 Find the product.

Simplify.

Check It Out! Example 2

Find the product.

B.

= 1 8

Evaluating Algebraic Expressions

4-1 Exponents

D. –94

= 25

–94 = –(9 • 9 • 9 • 9)

= –6,561

= (–5) • (–5) (–5)2

C. (–5)2

Evaluate.

Find the product.

Find the product. Then make the answer negative.

Check It Out! Example 2

Evaluating Algebraic Expressions

4-1 ExponentsAdditional Example 3: Using the Order of Operations

4(7) + 16

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

Simplify the powers.

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

Evaluating Algebraic Expressions

4-1 ExponentsCheck It Out! Example 3

60 – 7(7)

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

Simplify the powers.

Subtract inside the parentheses.

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

Evaluating Algebraic Expressions

4-1 Exponents

(72 – 3 • 7)1

2

Additional Example 4: Geometry Application

Simplify inside the parentheses.

Multiply.

Substitute the number of sides for n.

Subtract inside the parentheses.

14 diagonals

(49 – 21)1

2

(n2 – 3n)1

2

(28)1

2

Use the expression (n2 – 3n) to find the number of diagonals in a 7-sided figure.

1 2

Evaluating Algebraic Expressions

4-1 Exponents

A 7-sided figure has 14 diagonals. You can verify your answer by sketching the diagonals.

Additional Example 4 Continued

Evaluating Algebraic Expressions

4-1 Exponents

(42 – 3 • 4)1

2

Check It Out! Example 4

Simplify inside the parentheses.

Multiply

Substitute the number of sides for n.

Subtract inside the parentheses.

2 diagonals

(16 – 12)1

2

(n2 – 3n)1

2

(4)1

2

Use the expression (n2 – 3n) to find the number of diagonals in a 4-sided figure.

1 2

Evaluating Algebraic Expressions

4-1 Exponents

A 4-sided figure has 2 diagonals. You can verify your answer by sketching the diagonals.

Check It Out! Example 4 Continued

Evaluating Algebraic Expressions

4-1 ExponentsLesson Quiz

Write in exponential form.

1. n • n • n • n 2. (–8) • (–8) • (–8) • (h)

256

–213

(–8)3h

3. (–4)4

5. Evaluate xz – yx for x = 5, y = 3, and z = 6.

n4

6. A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15 25. How many are there after 5 minutes?480

Simplify.

4.