§ 7.2 Rational Exponents. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2 Rational...

§ 7.2

Rational Exponents

Blitzer, Intermediate Algebra, 5e – Slide #2 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.


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


Rational Exponents

EXAMPLEEXAMPLE

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

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

12

1

5

14 xy

SOLUTIONSOLUTION

5 45

14 33(a) xyxy

10100100(b) 2

1

46464(c) 33

1


Rational Exponents

EXAMPLEEXAMPLE

Rewrite with rational exponents:

.(b)13(a) 55 xx

SOLUTIONSOLUTION

5

15 1313(a) xx

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

2

5

2

15

2

155(b) xxxx


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

n

m


Rational Exponents

EXAMPLEEXAMPLE

Use radical notation to rewrite each expression and simplify:

.27(b)25(a) 3

22

3

SOLUTIONSOLUTION

12552525(a) 332

3

932727(b) 223

3

2


Rational Exponents

EXAMPLEEXAMPLE

Rewrite with rational exponents:

.11(b)(a)37 4 xyx

SOLUTIONSOLUTION

7

47 4(a) xx

2

331111(b) xyxy


Rational Exponents

The Definition of TIf is a nonzero real number, then

nma /

nma /

.1

//

nmnm

aa


Rational Exponents in Application

EXAMPLEEXAMPLE

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?

3

1

29xxf


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).

SOLUTIONSOLUTION

3

1

29xxf

CONTINUECONTINUEDD

This is the given formula.

3

1

272927 f Replace x with 27.

3 272927 f Rewrite as .

32927 f Evaluate the cube root.

3

1

27 3 27

8727 f Multiply.

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


Rational Exponents

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

bb

b


Rational Exponents

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.

CONTINUECONTINUEDD

mnnm bb

nnn baab

n

nn

b

a

b

a


Rational Exponents

EXAMPLEEXAMPLE

Simplify: .

5

55(c)(b)(a)

4

1

2

1

4

33

1

5

2

4

17

3

7

1

yx

x

x

SOLUTIONSOLUTION

7

2

7

1

7

37

3

7

1(a)

x

x

x

x

To divide with the same base,

subtract exponents.

Subtract.


Rational Exponents

15

2

12

1

15

2

12

1

3

1

5

23

1

4

13

1

5

2

4

1

(b)

y

x

yx

yxyx

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

Multiply:

CONTINUECONTINUEDD

5555

5

5

5

5

5

5

5

55(c) 14

4

4

1

4

5

4

1

4

5

4

1

4

2

4

3

4

1

2

1

4

3

4

1

2

1

4

3

.15

2

3

1

5

2 and

12

1

3

1

4

1

Rewrite with positive exponents.


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.


Rational Exponents

EXAMPLEEXAMPLE

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

SOLUTIONSOLUTION

Rewrite as exponential expressions.

Raise each factor in parentheses to its related power.

3

126

123 26 2(a) baabbaab

3

1

3

126

126

1

baba

3

1

3

2

6

2

6

1

baba

3

1

3

1

6

4

6

1

bbaa

3

1

3

1

6

4

6

1

ba

To raise powers to powers, multiply.

Reorder the factors.

To multiply with the same base, add exponents.


Rational Exponents

53

15 3 22(b) xx

Add.3

2

6

5

ba

Rewrite exponents with common denominators.

6

4

6

5

ba

Factor 1/6 out of the exponents. 6

145ba

Rewrite in radical notation.6 45ba

5

1

3

1

2

x

15

1

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.

CONTINUECONTINUEDD


Rational Exponents

Important to Remember:

Properties of integer exponents are true for rational exponents.Remember those exponent rules we learned before? Well, they still hold,

even when the exponents are fractions.

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.


Rational Exponents

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

Didn’t learn any radical rules? Then rewrite your expressionwithout the radical sign using rational exponents instead. Now you can use your exponent rules to simplify the radical expression.

Important to Remember:

§ 7.2 Rational Exponents. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2 Rational...

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