Chapter 1Chapter 1Section 3Section 3

Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc.

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22

33

Exponents, Roots, and Order of Operations

Use exponents.Find square roots.Use the order of operations.Evaluate algebraic expressions for given values of variables.

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1.31.31.31.3

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Objective 1

Use exponents.

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In algebra we use exponents as a way of writing products of repeated factors.

The number 5 shows that 2 is used as a factor 5 times.

The number 5 is the exponent, and 2 is the base.

25

5

5 factors o 2f

2 2 2 2 2 2

Exponent

Base

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Exponential ExpressionIf a is a real number and n is a natural number, then

an = a a a … a,

where n is the exponent, a is the base, and an is an exponential expression. Exponents are also called powers.

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n factors of a

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EXAMPLE 1

Write using exponents.

a.

Here, is used as a factor 4 times.

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2 2 2 2

7 7 7 7

2

74

2 2 2 2 2

7 7 7 7 7

4 factors of 2/7.

Read as “2/7 to the fourth power.”

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continued

b. (10)(–10)(–10)

c.

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= (10)3

Read as “10 cubed.”

y y y y y y y y 8y

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EXAMPLE 2

Evaluate 34.

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3 3 3 3 81 3 is used as a factor 4 times.

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EXAMPLE 3

Evaluate.

a. (3)2

b. 32

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( 3)( 3) 9 The base is 3.

(3 3) 9There are no parentheses. The exponent 2 applies only to the number 3, not to 3.

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The product of an odd number of negative factors is negative.

The product of an even number of negative factors is positive.

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CAUTION It is important to distinguish between an and (a)n.

Be careful when evaluating an exponential expression with a negative sign.

( ... )1n a a aa a

( )( ) ... ( )n

a a aa

The base is a.

The base is a.

n factors of a

n factors of a

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Objective 2

Find square roots.

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The opposite (inverse) of squaring a number is called taking its square root.

The square root of 49 is 7.

Another square root of 49 is 7, since (7)2 = 49.

Thus 49 has two square roots: 7 and 7.

We write the positive or principal square root of a number with the symbol , called a radical sign.

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The negative square root of 49 is written

Since the square of any nonzero real number is positive, the square root of a negative number, such as is not a real number.

49 7.

49

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EXAMPLE 4

Find each square root that is a real number.

a. b.

c. d.

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121

81 11

9 49 7

49 7 49

Not a real number, because the negative sign is inside the radical sign. No real number squared equals 49.

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Objective 3

Use the order of operations.

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Order of Operations

1. Work separately above and below any fraction bar.

2. If grouping symbols such as parentheses ( ), brackets [ ], or absolute value bars | | are present, start with the innermost set and work outward.

3. Evaluate all powers, roots, and absolute values.

4. Multiply or divide in order from left to right.

5. Add or subtract in order from left to right.

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NOTE Some students like to use the mnemonic “Please Excuse My Dear Aunt Sally” to help remember the rules for order of operations.

Be sure to multiply or divide in order from left to right; then add or subtract in order from left to right.

Please Excuse My Dear Aunt Sally

Parentheses Exponents Multiply Divide Add Subtract

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EXAMPLE 5

Simplify. a. 5 9 + 2 4

= 45 + 2 4 Multiply.

= 45 + 8 Multiply.

= 53 Add.

b. 4 – 12 4 2= 4 – 3 2 Divide.

= 4 – 6 Multiply.

= –2 Subtract.

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EXAMPLE 6

Simplify. (4 + 2) – 32 – (8 – 3)

(4 + 2) – 32 – (8 – 3)= 6 – 32 – 5 Perform operations inside parentheses.

= 6 – 9 – 5 Evaluate the power.

= –3 – 5 Subtract 6 – 9. = –8 Subtract.

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32 = 3 ·3 not 3 · 2

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EXAMPLE 7

Simplify.

Work separately above and below the fraction bar.

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2

110 6 9

25

12 3(2)6

2(2

1 110 6 10 6

2 25 5

12 3 12 3(4)

9

6)

6

3

Evaluate the root and the power.

5 6 3

10 3(4)

Multiply the fraction and whole number.

2

10 12

2

12

Subtract and add in the numerator.

Multiply 3(4).

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Objective 4

Evaluate algebraic expressions for given values of variables.

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Any sequence of numbers, variables, operation symbols, and/or grouping symbols formed in accordance with the rules of algebra is called an algebraic expression.

6ab, 5m – 9n, and 2(x2 + 4y)

We evaluate algebraic expressions by substituting given values for the variables.

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EXAMPLE 8

Evaluate the expression if w = 4, x = –12, y = 64 and z = –3.

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5( ) ( )

1

41

12

32 6

Substitute x = – 12, y = 64 and z = – 3.

Work separately above and below the fraction bar.

360 ( )(8)

13

60 24

13

84 84

13 13

5

1

x z y

x

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continued

Evaluate the expression w2 + 2z3 if w = 4, x = –12,

y = 64 and z = –3.

w2 + 2z3 = (4)2 + 2(–3)3

= 16 + 2(–27)

= 16 – 54

= –38

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Substitute w = 4 and z = – 3.

Evaluate the powers.

Multiply.

Subtract.