Section 10.1

Radical Expressions and

Functions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Radical Notation

• The Square Root Function

• The Cube Root Function

Radical Notation

Every positive number a has two square roots, one positive and one negative. Recall that the positive square root is called the principal square root.

The symbol is called the radical sign.

The expression under the radical sign is called the radicand, and an expression containing a radical sign is called a radical expression.

Examples of radical expressions:5

7, 6 2, and 3 4

xx

x

Example

Evaluate each square root.

a.

b.

c.

36

0.64

4

5

0.8

16

25

6

Example

Approximate to the nearest thousandth.

Solution

38

6.164

Example

Evaluate the cube root.

a.

b.

c.

3 64

3 125

1

2

5

31

8

4

Example

Find each root, if possible.

a. b. c.

Solution

a.

b.

c.

4 256 5 243 4 1296

4 256

5 243

4 1296

4 because 4 4 4 4 256.

53 because ( 3) 243.

An even root of a negative number is not a real number.

Example

Write each expression in terms of an absolute value.

a. b. c.

Solutiona.

b.

c.

2( 5) 2( 3)x 2 6 9w w

2( 5)

2( 3)x

2 6 9w w

5 5

3x

2( 3)w 3w

Example

If possible, evaluate f(1) and f(2) for each f(x).a. b.

Solutiona. b.

( ) 5 1f x x 2( ) 4f x x

(1) 5(1) 1

6

f

( 2) 5( 2) 1

9 undefined

f

2(1) 1 4

5

f

2( 2) ( 2) 4

8

f

( ) 5 1f x x 2( ) 4f x x

Example

Calculate the hang time for a ball that is kicked 75 feet into the air. Does the hang time double when a ball is kicked twice as high? Use the formula

SolutionThe hang time is

The hang time is

The hang times is less than double.

1( )

2T x x

1(75) 75

2T 4.3 sec

1(150) 150

2T 6.1 sec

Example

Find the domain of each function. Write your answer in interval notation.a. b.

SolutionSolve 3 – 4x 0.

The domain is

( ) 3 4f x x 2( ) 4f x x

3 4 0

4 3

3

4

x

x

x

3, .4

b. Regardless of the value of x; the expression is always positive. The function is defined for all real numbers, and it domain is , .

Section 10.2

Rational Exponents

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Basic Concepts

• Properties of Rational Exponents

Example

Write each expression in radical notation. Then evaluate the expression and round to the nearest hundredth when appropriate. a. b. c.

Solutiona. b.

c.

1/249 1/526 1/26x

1/249 49 7 1/526

1/2(6 )x 6x

Example

Write each expression in radical notation. Evaluate the expression by hand when possible.a. b.

Solutiona.

b.

2/38 3/410

2/38 2

3 8 22 4

3/410 34 10 4 1000

Example

Write each expression in radical notation. Evaluate the expression by hand when possible. a. b.Solutiona.

3/481 4/514

/4381

43/(81)

34 81

Take the fourth root of 81 and then cube it.

33

27

b. 4/514 Take the fifth root of 14 and then fourth it.

54/14

45 14

Cannot be evaluated by hand.

Example

Write each expression in radical notation and then evaluate. a. b.

Solutiona. b.

1/481 2/364

1/481 2/3641/4

1

81

4

1

81

1

3

3/2

1

64

23

1

64

2

1

4

1

16

Example

Use rational exponents to write each radical expression. a.

b.

c.

d.

7 3x

3

1

b3/2b

3/7x

25 ( 1)x 2/5( 1) x

2 24 a b2 2 1/4( ) a b

Example

Write each expression using rational exponents and simplify. Write the answer with a positive exponent. Assume that all variables are positive numbers.

a. b.4x x 34 256x1/2 1/4x x

1/2 1/4x

3/4x

1/43(256 )x

1/4 3 1/4256 ( )x

3/44x

Example (cont)

Write each expression using rational exponents and simplify. Write the answer with a positive exponent. Assume that all variables are positive numbers.

c. d.

5

4

32x

x

1/33

27

x

1/5

1/4

(32 )x

x

1/5 1/5

1/4

32 x

x

1/5 1/42x 1/202x

1/20

2

x

1/3

3

27

x

1/3

3 1/3

27

( )x

3

x

Example

Write each expression with positive rational exponents and simplify, if possible.a. b.

Solutiona.

b.

4 2x 1/4

1/5

y

x

1/41/2( 2)x

1/8( 2)x

1/5

1/4

x

y

4 2x

1/4

1/5

y

x

Section 10.3

Simplifying Radical

Expressions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Product Rule for Radical Expressions

• Quotient Rule for Radical Expressions

Product Rule for Radicals

Consider the following example:

Note: the product rule only works when the radicals have the same index.

4 25 2 5 10

4 25 100 10

Example

Multiply each radical expression.

a.

b.

c.

36 4

3 38 27

4 41 1 1 1 1

4 16 4 256 4

3 38 27 216 6

4 441 1 1

4 16 4

36 4 144 12

Example

Multiply each radical expression.

a.

b.

c.

2 4x x

3 23 5 10a a

44 4

3 7 2121

x y xy

y x xy

3 32 3 35 10 50 50a a a a

443 7x y

y x

2 4 6 3x x x x

Example

Simplify each expression. a.

b.

c.

500

3 40

72

100 5 10 5

3 3 38 5 2 5

36 2 6 2

Example

Simplify each expression. Assume that all variables are positive.a. b.

c.

449x575y

3 23 3 9a a w

4 249 7x x 4 325y y

4 325y y

25 3yy3 23 9a a w

3 327a w

3 33 27a w

33a w

Example

Simplify each expression.

a. b.37 7 3 5a a1/2 1/37 7 1/3 1/5a 1/2 1/37

5/67 8/15a

1/3 1/5a a

Quotient Rule

Consider the following examples of dividing radical expressions: 4 2 2 2

9 3 3 3

4 4 2

9 39

Example

Simplify each radical expression. Assume that all variables are positive. a. b.3

7

275

32

x3

3

7

27

3 7

3

5

5 32

x

5

2

x

Example

Simplify each radical expression. Assume that all variables are positive.

a. b.90

10

4x y

y

90

10

9

4x y

y

4x

32x

Example

Simplify the radical expression. Assume that all variables are positive.

4

55

32x

y

5 4

55

32x

y

5 45

55

32 x

y

5 42 x

y

Example

Simplify the expression.

Solution

1 1 x x

1 1 x x ( 1)( 1) x x

2 1 x

Example

Simplify the expression.

Solution

3 2

3

5 6

2

x x

x

3 2

3

5 6

2

x x

x3

( 3)( )

)2(

2

x

x

x

3 3 x

Section 10.4

Operations on Radical

Expressions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Addition and Subtraction

• Multiplication

• Rationalizing the Denominator

Radicals

Like radicals have the same index and the same radicand.

Like Unlike

3 2 5 2 3 2 5 3

Example

If possible, add the expressions and simplify.

a.

b.

c.

d.

4 7 8 7

3 37 5 2 5

6 16

39 5

8 13

12 7

The terms cannot be added because they are not like radicals.

The expression contains unlike radicals and cannot be added.

Example

Write each pair of terms as like radicals, if possible.a. b.Solution

a. b.

80, 1253 34 16, 7 54

125 25 5 5 5

80 16 5 4 5 33 34 16 4 8 2

33 37 54 7 27 2

3 34 2 2 8 2

3 37 3 2 21 2

Example

Add the expressions and simplify.

a. b.Solutiona. b.

20 5 5 5 2 50 72

4 5 5 5

20 5 5 5 2 50 72

=5 2 5 2 6 2

5 2 25 2 36 2

16 2

2 55 5

57

Example

Simplify the expressions.

a.

b.

8 7 2 7

3 3 37 5 2 5 5 3(7 2 1) 5

6 7

36 5

Example

Subtract and simplify. Assume that all variables are positive. a. b.5 549x x

33

77

64 4

yy

4 449x x x x

2 27x x x x

26x x

0

3 37 7

4 4

y y

Example

Subtract and simplify. Assume that all variables are positive. a. b.7 2 3 2

5 3

3 7 4 3343 24 27a b ab

7 2 3 3 2 5

5 3 3 5

21 2 15 2

15 15

6 2 2 2

15 5

343a6b33 ab3 24 273 ab3

7a2b ab3 243 ab3

7a2b ab3 72 ab3

(7a2b 72) ab3

Example

Multiply and simplify.

Solution

3 5x x

3 5x x 3 5 3 5x x x x

215 3 5x x x

15 2 x x

Example

Rationalize each denominator. Assume that all variables are positive.a. b. c.

Solutiona.

b.

c.

1

3

7

8 35

ab

b

1

3

3

3

3

3

7

8 33

3 7 3

8 3

7 3

24

5

ab

b2

ab

b b

2

ab b

b b b

2

ab b

b b

2

a b

b

Conjugates

Example

Rationalize the denominator of

Solution

1.

1 3

1

1 31 3

1 33 1

1

2 2

1 3

1 ( 3)

1 3

1 3

1 3

2

1 3

2 2

1 3

2 2

Example

Rationalize the denominator of

Solution

4 6

3 6

3 6

3 6

4 6

3 6

2

2

12 4 6 3 6 ( 6)

9 ( 6)

18 7 6

3

18 7 6

3 3

7 66

3

4 6

3 6

Example

Rationalize the denominator of

Solution

3 2

4

x

2/3

4

x

1/3

2/3 1/3

4 x

x x

1/3

2/3 1/3

4x

x

34 x

x

3 2

4

x

Section 10.6

Equations Involving Radical

Expressions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Solving Radical Equations

• The Distance Formula

• Solving the Equation xn = k

If each side of an equation is raised to the same positive integer power, then any solutions to the given equations are among the solutions to the new equation. That is, the solutions to the equation a = b are among the solutions to an = bn.

POWER RULE FOR SOLVING EQUATIONS

Example

Solve Check your solution.

Solution

4 2 1 12. x

4 2 1 12x

2 1 3x

222 1 3x

2 1 9x 2 8x

4x

Check:

4 2 4 1 12

4 9 12 4 3 1212 12

It checks.

Step 1: Isolate a radical term on one side of the equation.

Step 2: Apply the power rule by raising each side of the equation to the power equal to the index of the isolated radical term.

Step 3: Solve the equation. If it still contains radical, repeat Steps 1 and 2.

Step 4: Check your answers by substituting each result in the given equation.

SOLVING A RADICAL EQUATION

Example

Solve

SolutionStep 1: To isolate the radical term, we add 3 to each side

of the equation.

Step 2: Square each side.

Step 3: Solve the resulting equation.

6 3 1.x

6 4x

6 3 1x

226 4x

6 16x 10x

10x

Example (cont)

Step 4: Check your answer by substituting into the given equation.

Since this checks, the solution is x = −10.

6 3 1.x

6 3 1x

6 10 3 1

16 3 1

4 3 1

1 1

Example

Solve Check your results and then solve the equation graphically. SolutionSymbolic Solution

3 2 2. x x

3 2 2x x

2 2

3 2 2x x 23 2 4 4x x x 20 7 6x x

0 6 1x x

6 or = 1x x

Check:3 2 2x x

3 6 2 6 2

4 4

3 2 2x x

3 1 2 1 2

1 1

It checks.

Thus 1 is an extraneous solution.

Example (cont)

Graphical Solution

The solution 6 is supported graphically where the intersection is at (6, 4). The graphical solution does not give an extraneous solution.

3 2 2 x x

Example

SolveSolution

4 3 3x x

4 3 3x x

2 2

4 3 3x x

224 2 4 (3) 3 3x x x

4 6 4 9 3x x x

3 12 6 4x x

223 12 6 4x x

29 72 144 36(4 )x x x

29 72 144 144x x x 29 72 144 0x x

2(3 12) 0x

3 12 0x 3 12x

4x

The answer checks. The solution is 4.

Example

Solve.

SolutionStep 1: The cube root is already isolated, so we proceed

to Step 2.Step 2: Cube each side.Step 3: Solve the resulting equation.

3 2 6 4 x

333 2 6 4x

2 6 64x

2 58x

29x

Example (cont)

Step 4: Check the answer by substituting into the given equation.

3 2 6 4 x

3 2 6 4x

3 2 29 6 4 3 64 4

4 4

Since this checks, the solution is x = 29.

Example

Solve x3/4 = 4 – x2 graphically. This equation would be difficult to solve symbolically, but an approximate solution can be found graphically.Solution

Example

A 6ft ladder is placed against a garage with its base 3 ft from the building. How high above the ground is the top of the ladder? Solution

2 2 2c a b 2 2 26 3a

236 9a 227 a

27 a3 3 a The ladder is 5.2 ft above ground.

The distance d between the points (x1, y1) and (x2, y2) in the xy-plane is

DISTANCE FORMULA

2 2

2 1 2 1 .d x x y y

Example

Find the distance between the points (−1, 2) and (6, 4).Solution

2 2

2 1 2 1 .d x x y y

2 26 1 4 2d

49 4d

53d

7.28d

Take the nth root of each side of xn = k to obtain

1. If n is odd, then and the equation becomes

2. If n is even and k > 0, then and the equation becomes

(If k < 0, there are no real solutions.)

SOLVING THE EQUATION xn = k

.n n nx k

n nx x.nx k

.nx k

n nx x

Example

Solve each equation.a. x3 = −216 b. x2 = 17 c. 3(x + 4)4 = 48Solutiona. b.

3 3 3 216x

6x

3 216x

or

2 17x

17x

17x

2 17x

Example (cont)

c. 3(x + 4)4 = 48

44 4 16x

4 2x

4 2x 4 2x

6x 2x

4( 4) 16x

Example

The formula for the volume (V) of a sphere with a radius (r), is given by Solve for r.

Solution

34.

3V r

34

3V r

33

4

Vr

33

4

Vr

Section 10.7

Complex Numbers

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Basic Concepts

• Addition, Subtraction, and Multiplication

• Powers of i

• Complex Conjugates and Division

PROPERTIES OF THE IMAGINARY UNIT i

21 and 1i i

THE EXPRESSION a

If > 0, then = . a a i a

Example

Write each square root using the imaginary i.a. b. c.

Solution

a.

b.

c.

36 15 45

36 36i 6i

15 15i

45 45i 9 5i 3 5i

Let a + bi and c + di be two complex numbers. Then

(a + bi) + (c+ di) = (a + c) + (b + d)i Sum

and

(a + bi) − (c+ di) = (a − c) + (b − d)i. Difference

SUM OR DIFFERENCE OF COMPLEX NUMBERS

Example

Write each sum or difference in standard form.a. (−8 + 2i) + (5 + 6i) b. 9i – (3 – 2i)Solutiona. (−8 + 2i) + (5 + 6i)

= (−8 + 5) + (2 + 6)I= −3 + 8i

b. 9i – (3 – 2i)= 9i – 3 + 2i= – 3 + (9 + 2)I= – 3 + 11i

Example

Write each product in standard form.a. (6 − 3i)(2 + 2i) b. (6 + 7i)(6 – 7i)Solutiona. (6 − 3i)(2 + 2i)

= (6)(2) + (6)(2i) – (2)(3i) – (3i)(2i)

= 12 + 12i – 6i – 6i2

= 12 + 12i – 6i – 6(−1)

= 18 + 6i

Example (cont)

Write each product in standard form.a. (6 − 3i)(2 + 2i) b. (6 + 7i)(6 – 7i)Solutionb. (6 + 7i)(6 – 7i)

= (6)(6) − (6)(7i) + (6)(7i) − (7i)(7i)

= 36 − 42i + 42i − 49i2

= 36 − 49i2

= 36 − 49(−1)

= 85

The value of in can be found by dividing n (a positive integer) by 4. If the remainder is r, then

in = ir.

Note that i0 = 1, i1 = i, i2 = −1, and i3 = −i.

POWERS OF i

Example

Evaluate each expression.a. i25 b. i7 c. i44

Solutiona. When 25 is divided by 4, the result is 6 with the remainder of 1. Thus i25 = i1 = i.

b. When 7 is divided by 4, the result is 1 with the remainder of 3. Thus i7 = i3 = −i.

c. When 44 is divided by 4, the result is 11 with the remainder of 0. Thus i44 = i0 = 1.

Example

Write each quotient in standard form.a. b.

Solutiona.

3 2

5

i

i

9

3i

3 2

5

i

i

3 2 5

5 5

i i

i i

3 5 3 2 5 2

5 5 5 5

i i i i

i i i i

2

2

15 3 10 2

25 5 5

i i i

i i i

15 7 2 1

25 1

i

17 7

26

i

17 7

26 26

i

Example (cont)

b. 9

3i

9 3

3 3

i

i i

2

27

9

i

i

27

9 1

i

27

9

i

3i