§ 7.2

Rational Exponents

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2

Rational Exponents

The Definition of TIf represents a real number and is an integer, then

If a is negative, n must be odd. If a is nonnegative, n can be any index.

n a

na /1

2n

./1 nn aa

P 499

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.2

Rational ExponentsEXAMPLE

Use radical notation to rewrite each expression. Simplify, if possible:

.64(c)100(b)3(a) 31

21

51

4 xy

SOLUTION

5 451

4 33(a) xyxy

10100100(b) 21

46464(c) 331

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.2

Rational ExponentsCheck Point 1 on page 499

21

25(a)

31

8(b)

525

283

41

25(c) xy 4 25xy

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.2

Rational ExponentsEXAMPLE

Rewrite with rational exponents:

.(b)13(a) 55 xx

SOLUTION

51

5 1313(a) xx

Parentheses are needed in part (a) to show that the entire radicand becomes the base.

25

215

21

55(b) xxxx

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.2

Rational ExponentsCheck Point 2 on p 500

4 5(a) xy 41

5xy

53

2(b) ba 5

13

2

ba

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.2

Rational Exponents

The Definition of TIf represents a real number, is a positive rational number reduced to lowest terms, and is an integer, then

and

n a

nma /

2n

mnnm aa /

./ n mnm aa

nm

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.2

Rational ExponentsEXAMPLE

Use radical notation to rewrite each expression and simplify:

.27(b)25(a) 32

23

SOLUTION

12552525(a) 3323

932727(b) 22332

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.2

Rational ExponentsCheck Point 3 on p 501

43

81-(c)

34

8(a) 43 8 16

34 81 27

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.2

Rational ExponentsEXAMPLE

Rewrite with rational exponents:

.11(b)(a)37 4 xyx

SOLUTION

74

7 4(a) xx

233

1111(b) xyxy

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.2

Rational ExponentsCheck Point 4 on p 501

3 46(a) 34

6

75 2(b) xy 57

2xy

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.2

Rational Exponents

The Definition of TIf is a nonzero real number, then

nma /

nma /

.1/

/nm

nm

aa

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.2

Rational ExponentsCheck Point 5 on p 502

21

100(a)

101

21

100

1

31

8(b)

21

31

8

1

53

32(c)

81

53

32

1

95

3(d) xy 9

53

1

xy

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.2

Rational Exponents in ApplicationEXAMPLE

The Galapagos Islands, lying 600 miles west of Ecuador, are famed for their extraordinary wildlife. The function

models the number of plant species, f (x), on the various islands of the Galapagos chain in terms of the area, x, in square miles, of a particular island. Use the function to solve the following problem.

How many species of plants are on a Galapagos island that has an area of 27 square miles?

31

29xxf

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.2

Rational Exponents in Application

Because we are interested in how many species of plants there are on a Galapagos island having an area of 27 square miles, substitute 27 for x. Then calculate f (x).

SOLUTION

31

29xxf

CONTINUED

This is the given formula.

31

272927 f Replace x with 27.

3 272927 f Rewrite as .

32927 f Evaluate the cube root.

31

27 3 27

8727 f Multiply.A Galapagos island having an area of 27 square miles contains approximately 87 plant species.

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.2

Rational Exponents p 502

Properties of Rational ExponentsIf m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then

1) When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

2) When dividing exponential expressions with the same base, subtract the exponents. Use this difference as the exponent of the common base.

nmnm bbb

nmn

m

bbb

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.2

Rational Exponents p 502

Properties of Rational ExponentsIf m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then

3) When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

4) When a product (not sum) is raised to a power, raise each factor to that power and multiply.

5) When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

CONTINUED

mnnm bb

nnn baab

n

nn

ba

ba

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.2

Rational ExponentsEXAMPLE

Simplify: .5

55(c)(b)(a)41

21

43

31

52

417

3

71

yx

x

x

SOLUTION

72

71

737

3

71(a)

x

xx

x

To divide with the same base,

subtract exponents.

Subtract.

Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.2

Rational Exponents

152

121

152

121

31

523

1

413

1

52

41

(b)

y

x

yx

yxyx

To raise a product to a power, raise each factor to the power.

Multiply:

CONTINUED

55555

5

5

5

5

5

5

55(c) 144

41

45

41

45

41

42

43

41

21

43

41

21

43

.152

31

52 and

121

31

41

Rewrite with positive exponents.

Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.2

Rational ExponentsCheck Point 6 on page 503Simplify:

43

52

1.9(c)

34

10

50(b)31

x

x

31

21

77(a) 31

21

7

62

63

7

65

7

34

31

5

x15 x x

5

20

6

1.9 103

1.9

31

41

53

(d)

yx

121

153

yx51

121

x

y

Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.2

Rational Exponents

Simplifying Radical Expressions Using Rational Exponents

1) Rewrite each radical expression as an exponential expression with a rational exponent.2) Simplify using properties of rational exponents.3) Rewrite in radical notation if rational exponents still appear.

Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.2

Rational ExponentsEXAMPLE

Use rational exponents to simplify: .2(b)(a) 5 33 26 2 xbaab

SOLUTIONRewrite as exponential expressions.Raise each factor in parentheses to its related power.

31

261

23 26 2(a) baabbaab

31

31

261

261

baba

31

32

62

61

baba

31

31

64

61

bbaa

31

31

64

61

ba

To raise powers to powers, multiply.Reorder the factors.

To multiply with the same base, add exponents.

Blitzer, Intermediate Algebra, 5e – Slide #23 Section 7.2

Rational Exponents

531

5 3 22(b) xx

Add.32

65

ba

Rewrite exponents with common denominators.

64

65

ba

Factor 1/6 out of the exponents. 61

45ba

Rewrite in radical notation.6 45ba

51

31

2

x

151

2x

Write the radicand as an exponential expression.Write the entire expression in exponential form.To raise powers to powers, multiply the exponents.

15 2x Rewrite in radical notation.

CONTINUED

Blitzer, Intermediate Algebra, 5e – Slide #24 Section 7.2

Rational Exponents

Important to Remember:

• An expression with rational exponents is simplified when no parentheses appear,

no powers are raised to powers, each base occurs once, and

no negative or zero exponents appear.

• Some radical expressions can be simplified using rational exponents. Rewrite the expression using rational exponents, simplify, and rewrite in radical notation if rational exponents still appear.

DONE

Blitzer, Intermediate Algebra, 5e – Slide #26 Section 7.2

Rational Exponents

Rational exponents have been defined in such a way so as to make their properties the same as the properties for integer exponents.

In this section we explore the meaning of a base raised to a rational (fractional) exponent.

We will also discover how we can use rational exponents to simplify radical expressions.

Blitzer, Intermediate Algebra, 5e – Slide #27 Section 7.2

Rational Exponents

Important to Remember: