Chapter 7

Radical Functions and Rational Exponents

In this chapter, you will …

You will extend your knowledge of roots to include cube roots, fourth roots, fifth roots, and so on.

You will learn to add, subtract, multiply, and divide radical expressions, including binomial radical expressions.

You will solve radical equations, and graph translations of radical functions and their inverses.

7-1 Roots and Radical Expressions

What you’ll learn … To simplify nth roots

1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

Definition nth Root

For any real numbers a and b, any positive integer n, if an = b, then a is an nth root of b.

Since 52 = 25, 5 is a square root of 25.

Since 53 = 125, 5 is a cube root of 125.

Since 54 = 625, 5 is a fourth root of 625.

Since 55 = 3125, 5 is a fifth root of 3125.

This pattern leads to the definition of nth root.

24 = 16 2 is 4th root of 16. (-2)4 = 16 -2 is 4th root of 16. x4 = -16 No 4th root of -16. (√10)4 = 100 4th root of 100 is√10.

Type of Number

Number of Real nth Roots when n

is Even

Number of Real nth Roots when n

is Odd

Positive 2 1

0 1 1

Negative None 1

Example 1a Finding All Real Roots

Find all cube roots of 0.008, -1000, and .

127

Find all square roots of .0001, -1 and . 36121

Example 1b Finding All Real Roots

Find all fifth roots of 0, -1, and 32 .

Find all fourth roots of -.0001, 1 and . 16 81

A radical sign is used to indicate a root.

The number under the radical sign is the radicand.

The index gives the degree of the root.

radical signradical sign

When a number has two real roots, the positive root is called the principal root and the radical sign indicates the principal root. The principal fourth root of 16 is written

The principal fourth root of 16 is 2 because

equals . The other fourth root of 16 is written as which equals -2.

√164

√16 √244 4

- √164

Example 2 Finding Roots

Find each real number root.

√-27

√81

√49

3

4

Notice that when x=5, √x2 = √52 = √25 = 5 =x.

And when x=-5, √x2 = √(-5)2 = √25 = 5 ≠ x.

Property nth Root of an, a < 0

For any negative real number a,

√an = a when n is even.n

Example 3a Simplifying Radical Expressions

Simplify each radical expression.

√4x6

√a3b6

√x4y8

3

4

Example 3b Simplifying Radical Expressions

Simplify each radical expression.

√4x2y4

√-27c6

√x8y12

3

4

Example 4 Real World Connection A citrus grower wants to ship a select

grade of oranges that weigh from 8 to 9 ounces in gift cartons. Each carton will hold three dozen oranges, in 3 layers of 3 oranges by 4 oranges.

The weight of an orange is related to its diameter by the formula w = , where d is the diameter in inches and w is the weight in ounces. Cartons can be ordered in whole inch dimensions. What size cartons should the grower order?

Find the diameter if w = 3 oz 5.5 oz 6.25 oz.

d3

4

7-2 Multiplying and Dividing Radical Expressions

What you’ll learn … To multiply radical expressions To divide radical expressions

1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

To multiply radicals consider the following:

√16 √9 = 4 3 =12 and √16 9 = √144 = 12** *

Property Multiplying Radical Expressions

If √a and √b are real numbers, then √a √b = √ab.

n n

nn n*

Example 1a Multiplying RadicalsMultiply. Simplify if possible.

√3 √12

√3 √-9

√4 √ -4

3

4

3

4

*

*

*

Example 2 Simplifying Radical Expressions

Simplify each expressions. Assume that all variables are positive.

• √50x4

• √18x4

• 3√7x3 2√21x3y2*

3

Example 3 Multiplying Radical Expressions

Multiply and simplify.

• 3√7x3 2√21x3y2

• √54x2y3 √5x3y4

*

3 3*

To divide radicals consider the following:

√36 6 and 36 (6)2 √36

√25 5 25 (5)2 √25

== =

Property Dividing Radical Expressions

If √a and √b are real numbers, then √a a

√b bn

n

n n

= n

Example 4 Dividing Radicals

Multiply. Simplify if possible.

√243 √12x4

√27 √3x

√1024x15

√4x

To rationalize a denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.

Example 5 Rationalizing the Denominator

Rationalize the denominator of each expression.

7 √2x3 √4 5 √10xy √6x

7-3 Binomial Radical Expressions

What you’ll learn … To add and subtract radical expressions To multiply and divide binomial radical

expressions

1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

Like radicals are radical expressions that have the same index and the same radicand. To add or subtract like radicals, use the Distributive Property.

Example 1 Adding and Subtracting Radical Expressions

5 √ x - 3 √ x 4 √ xy + 5 √ xy

4 √ 2 - 5 √ 3 7 √ 5 - 2 √5

2 √ 7 + 3 √ 7

3 3

4 3

Example 2 Simplifying Before Adding or Subtracting

6 √ 18 + 4 √ 8 - 3√ 72

√ 50 + 3 √ 32 - 5 √ 18

Example 4 Multiplying Binomial Radical Expressions

(3 + 2√ 5 ) ( 2 + 4 √ 5 )

(√ 2 - √ 5 )

2

Example 5 Multiplying Conjugates

(2 + √ 3 ) ( 2 - √ 3 )

(√ 2 - √ 5 ) (√ 2 + √ 5 )

Example 6 Rationalizing a Binomial Radical

Denominator

3 + √5

1 - √5

6 + √15

4 - √15

7-4 Rational Exponents

What you’ll learn … To simplify expressions with

rational exponents

1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

Another way to write a radical expression is to use a rational exponent.

Like the radical form, the exponent form always indicates the principal root.

√25 = 25½

√27 = 27⅓3

√16 = 161/44

Example 1 Simplifying Expressions with Rational Exponents

1251/3

5½

2½ 2½

2½ 8½

*

*

P/R = power/root

√x pr

( √x )pr

A rational exponent may have a numerator other than 1. The property (am)n = amn shows how to rewrite an expression with an exponent that is an improper fraction.

Example

253/2 = 25(3*1/2) = (253)½ = √253

Example 2 Converting to and from Radical Form

x3/5

y -2.5

y -3/8

√a3

( √b )2

√x2

5

3

Properties of Rational Exponents

Let m and n represent rational numbers. Assume that no denominator = 0.

Property Example

am * an = a m+n 8⅓ * 8⅔ = 8 ⅓+⅔ = 81 =8

(am)n = amn (5½)4 = 5½*4 = 52 = 25

(ab)m = ambm (4 *5)½ = 4½ * 5½ =2 * 5½

Properties of Rational Exponents

Let m and n represent rational numbers. Assume that no denominator = 0.

Property Example

a-m 1 9 -½ 1 1

am 9 ½ 3

am a m-n π3/2 π 3/2-1/2 = π1 = π

an π ½

a m am 5 5⅓ 5⅓

b bm 27 27 ⅓ 3

= ==

=

=

=

⅓

=

Example 4 Simplifying Numbers with Rational Exponents

(-32)3/5

4 -3.5

Example 5 Writing Expressions in Simplest Form

(16y-8) -3/4

(8x15)-1/3

7-5 Solving Radical Equations

What you’ll learn … To solve radical equations

2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent.

Radical Equation

Not a Radical Equation

Steps for Solving a Radical Equation

1. Get radical by itself.

2. Raise both sides to index power.

3. Solve for x.

4. Check.

Example 1 Solving Radical Equations with Index 2

Solve

2 + √3x-2 = 6

√5x+1 – 6 = 0

Example 2 Solving Radical Equations with Rational Exponents

Solve

2 (x – 2)2/3 = 50

3(x+1)3/5 = 24

Real World Connection

A company manufactures solar cells that produce 0.02 watts of power per square centimeter of surface area. A circular solar cell needs to produce at least 10 watts. What is the minimum radius?

Example 4 Checking for Extraneous Solutions

Solve

√x – 3 + 5 = x

√3x + 2 - √2x + 7 = 0

Example 5 Solving Equations with Two Rational Exponents

Solve

(2x +1)0.5 – (3x+4)0.25 = 0

Solve

(x +1)2/3 – (9x+1)1/3 = 0

7-8 Graphing Radical Functions

What you’ll learn … Graph radical functions

2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

A radical equation defines a radical function.

The graph of the radical function y= √x + k is

a translation of the graph of y= √x. If k is

positive, the graph is translated k units up. If

k is negative, the graph is translated k units

down.

Example 1 Translating Square Root Functions Vertically

y = √x y = √x + 3

Example 2 Translating Square Root Functions Horizontally

y = √x y = √x + 3

Example 3 Graphing Square Root Functions

y = -√x

Example 4 Graphing Square Root Functions

y = -2√x+1 - 3

Real World Connection

The function h(x) = 0.4 √ x models the height h in meters of a female giraffe that has a mass of x kilograms. Graph the model with a graphing calculator. Use the graph to estimate the mass of the young giraffe in the photo.

3

2.5 m

Example 6 Graphing Cube Root Functions

y = 2√x+3 - 1

3

Example 7 Transforming Radical Equations

Rewrite to make it easy to graph using a translation. Describe the graph.

y = √4x-12

Rewrite to make it easy to graph using a translation. Describe the graph.

y = √8x-24 +3

3

In this chapter, you should have …

Extended your knowledge of roots to include cube roots, fourth roots, fifth roots, and so on.

Learned to add, subtract, multiply, and divide radical expressions, including binomial radical expressions.

Solved radical equations, and graphed translations of radical functions and their inverses.