Section 9.1

Finding Roots

OBJECTIVES

Find the square root of a number.

A

Square a radical expression.

B

OBJECTIVES

Classify the square root of a number and approximate it with a calculator.

C

OBJECTIVES

Find higher roots of numbers.

D

Solve an application involving square roots.

E

DEFINITION

If a is a positive real number,

a = b is the positive square root of a so that b2 = a.

– a = b is the negative square root of a so that b2 = a.

Square Root

DEFINITION

If a = 0, a = 0 and 0 = 0since 02 = 0

Square Root

When the square root of a non-negative real number a is squared, the result is that positive real number:

RULESquaring a Square Root

2 2= and – = .a a a a

RULE

If a is negative, a isnot a real number.

Square Root of aNegative Number

Section 9.1Exercise #1

Chapter 9Roots and Radicals

Find.

a) 169 b) – 49

81

= 132

= 13

= – 72

92

= – 7

9

Section 9.1Exercise #2

Chapter 9Roots and Radicals

Find the square of each radical expression.

a) – 121 b) x2 + 7

2 = – 121

= 121

22 = + 7

x

= x 2 + 7

Section 9.1Exercise #3

Chapter 9Roots and Radicals

Classify each number as rational, irrational, or not a real number, and simplify if possible.

a) 17 b) – 36

irrational

Not real

= – 6

=

10

7

c) –100 d) 100

49

rational

rational

Section 9.1Exercise #4

Chapter 9Roots and Radicals

Find each root, if possible.

a) 814 b) – 6254

= 344 = – 544

= 3 = – 5

Section 9.1Exercise #5

Chapter 9Roots and Radicals

t =

d

5

d = 20 t =

205

t = 4 = 2

The diver takes 2 seconds.

A diver jumps from a cliff 20 meters high. If the time t (in seconds) it takes an object dropped from a distance

d (in meters) to reach the ground is given by: How long does it take the diver to reach the water?

Section 9.2

Multiplication and Division of Radicals

OBJECTIVES

Multiply and simplify radicals using the product rule.

A

OBJECTIVES

Divide and simplify radicals using the quotient rule.

B

OBJECTIVES

Simplify radicals involving variables.

C

OBJECTIVES

Simplify higher roots.D

If a and b are nonnegative numbers,

Product Rule for Radicals

= ba b a

If a and b are positive numbers,

Quotient Rule for Radicals

ab

= ab

For any real number a,

Absolute Value for Radicals

a2 = a

For all real numbers where the indicated roots exist,

Properties of Radicals

= and = nn nn nn

a aa b a bb b

Section 9.2Exercise #6

Chapter 9Roots and Radicals

Simplify.

a) 125 b) 54

= 25 • 5

= 5 5

= 9 • 6

= 3 6

Section 9.2Exercise #7

Chapter 9Roots and Radicals

Multiply.

a) 3 • 11 b) 11 • y , y > 0

= 3 • 11

= 33

= 11 • y

= 11y

Section 9.2Exercise #8

Chapter 9Roots and Radicals

Simplify.

a)

7

16 b)

21 50

7 5

=

7

16

=

7

4

=

21

7 •

50

5

= 3 10

Section 9.2Exercise #9

Chapter 9Roots and Radicals

Simplify.

a) 144n2 , n > 0 b) 32y7 , y > 0

= 122n2

= 12n

= 42 • 2 • y6 • y

= 4y3 2y

Section 9.2Exercise #10

Chapter 9Roots and Radicals

Simplify.

a) 964

= 16 • 64

= 24 • 64

= 2 64

Section 9.3

Addition and Subtractions of Radicals

OBJECTIVES

Add and subtract like radicals.

A

OBJECTIVES

Use the distributive property to simplify radicals.

B

OBJECTIVES

Rationalize the denominator in an expression.

C

Rationalizing Denominators

PROCEDURE

Method 1:

Multiply both numerator and denominator of the fraction by the square root in the denominator.

Rationalizing Denominators

PROCEDURE

Method 2: Multiply numerator and denominator by the square root of a number that makes the denominator the square root of a perfect square.

Section 9.3Exercise #11

Chapter 9Roots and Radicals

Simplify.

a) 9 13 + 7 13 b) 14 6 – 3 6

= 9 + 7 13

= 16 13

= 1 4 – 3 6

= 11 6

Section 9.3Exercise #12

Chapter 9Roots and Radicals

Simplify.

a) 28 + 63

= 4 7 + 9 7

= 2 7 + 3 7

= 2 + 3 7

= 5 7

Section 9.3Exercise #13

Chapter 9Roots and Radicals

Simplify.

3 a) 18 – 5 • = 3 18 – 3 • 5

= 3 • 9 • 2 – 3 • 5

= 9 • 3 • 2 – 3 5

= 3 6 – 15

Section 9.3Exercise #14

Chapter 9Roots and Radicals

=

3 • 5

20 • 5

Write

3

20 with a rationalized denominator.

=

15

100

=

15

10

Section 9.3Exercise #15

Chapter 9Roots and Radicals

=

y 2 • 2

50 • 2

Write

y 2

50, y > 0 with a rationalized denominator.

=

y 2

100

=

y 2

10

Section 9.4

Simplifying Radicals

OBJECTIVES

Simplify a radical expression involving products, quotients, sums, or differences.

A

OBJECTIVES

Use the conjugate of a number to rationalize the denominator of an expression.

B

OBJECTIVES

Reduce a fraction involving a radical by factoring.

C

Simplifying Radical ExpressionsRULES

1. Whenever possible, write the rational-number representation of a radical expression.

34 2 1 181 as 9, as , and as 9 3 8 2

Simplifying Radical ExpressionsRULES

2. Use the product rule x • y = xy to write indicated products as a single radical.

Simplifying Radical ExpressionsRULES

6 instead of 2 • 3 and 2ab instead of 2a • b

Section 9.4Exercise #16

Chapter 9Roots and Radicals

= 8 14 – 14 a) 8 14 – 7 • 2

Simplify.

= 8 – 1 14

= 7 14

Simplify.

b) 12x 3

4x 2, x > 0

=

12x 3

4x 2

=

12

4 • x3 – 2

= 3x

Section 9.4Exercise #17

Chapter 9Roots and Radicals

=

500

23

a)

5003

23

Simplify.

= 2503

= 125 • 23

= 5 23

Section 9.4Exercise #18

Chapter 9Roots and Radicals

10 – 2 20 10 + b) 2 20

Simplify.

2 2 = 10 – 2 20

= 10 – 4 20

= 10 – 80

= – 70

Section 9.4Exercise #19

Chapter 9Roots and Radicals

a)

11

3 + 1

Simplify.

3 – 111 •

=3 + 1 1 3 –

11 3 – 1 • =

3 – 1

=

11 3 – 11

2

Section 9.4Exercise #20

Chapter 9Roots and Radicals

a)

– 6 + 18

3

Simplify.

=

– 6 + 9 • 2

3

=

– 6 + 3 2

3

3 – 2 + 2 =

3

= – 2 + 2

Section 9.5

Applications

OBJECTIVES

Solve equations with one square root term containing the variable.

A

OBJECTIVES

Solve equations with two square root terms containing the variable.

B

OBJECTIVES

Solve an application.C

Raising Both Sides of an Equation to a Power

If both sides of the equation A = B

are squared, all solutions are

among the solutions of the new

equation A2 =B2.

PROCEDURE

Solving Radical EquationsPROCEDURE

1. Isolate the square root term containing the variable.

2. Square both sides of the equation.

Solving Radical EquationsPROCEDURE

3. Simplify and repeat steps 1 and 2 if there is a square root term containing the variable.

Solving Radical EquationsPROCEDURE

4. Solve the resulting linear or quadratic equation.

5. Check all proposed solutions in the original equation.

Section 9.5Exercise #22

Chapter 9Roots and Radicals

x + 4 – x = 2

Solve.

x + 4 = x + 2

x + 4 = x 2 + 4x + 4

0 = x 2 + 3x

0 = + 3x x

x = 0 or x = – 3

Solve.

x = 0 or x = – 3

2 – 0 = 2

4 2 1 + 3 ? 2

Check: x = – 3, – 3 + 4 – – 3 ? 2

Check: x = 0, 0 + 4 – 0 ? 2

Section 9.5Exercise #23

Chapter 9Roots and Radicals

y + 3 = 2y + 1

Solve.

y + 3 = 2y + 1

3 = y + 1

Check: 2 + 3 ? 2 • 2 + 1

5 ? 4 + 1

5 = 5

2 = y y = 2

Section 9.5Exercise #24

Chapter 9Roots and Radicals

y + 6 – 3 2y – 5 = 0Solve.

y + 6 = 3 2y – 5

y + 6 = 9 2y – 5 y + 6 = 18y – 45

6 = 17y – 45

51 = 17y

3 = y

3 + 6 – 3 2 • 3 – 5 ? 0

9 – 3 6 – 5 ? 0

0 = 0

3 – 3 1 ? 0

3 – 3 ? 0

Solve.

3 = y

Check:

y + 6 – 3 2y – 5 = 0