Cap07

Name:

Chapter 7 Pretest Form A

Date:

Evaluate. 1.

2.

5

121

1. ____________________________

− 32

2. ____________________________

3. Use absolute value to evaluate

4. Write

( − 5b )2

m 2 + 8m + 16 .

3. ____________________________

as an absolute value.

4. ____________________________

Write in exponential form. Assume all variables represent positive real numbers. 5.

6.

4

53

5. ____________________________

3a

6. ____________________________

Write in radical form. Assume all variables represent positive real numbers. 7.

( 3x )1 4

7. ____________________________

14

⎛1 ⎞ 8. ⎜ x 3 y ⎟ ⎝2 ⎠

8. ____________________________

Simplify. Assume all variables represent positive real numbers. 9.

5

3 32

10.

3

18x12 y 3

10. ____________________________

80c

11. ____________________________

11.

12.

13.

9. ____________________________

180ab 4

12. ____________________________

5ab 2

98 − 50 − 72

13. ____________________________

259

Chapter 7 Pretest Form A (cont.) 14.

15.

16.

17.

(

)(

2 +1

2 −3

)

14. ____________________________

3

15. ____________________________

3−2

(8 −

Name:

)(

− 1 − 2 + − 16

)

16. ____________________________

− 12

17. ____________________________

7 − −1

Solve. 18. 2 x = 5 x − 16

18. ____________________________

19. 2 + u = 2u + 7

19. ____________________________

20.

x +3= x−3

20. ____________________________

260

Name:

Chapter 7 Pretest Form B

Date:

Evaluate. 1.

2.

3

81

1. ____________________________

−1000

2. ____________________________

(−17)2 .

3. ____________________________

x 2 + 6 x + 9 as an absolute value.

4. ____________________________

3. Use absolute value to evaluate

4. Write

Write in exponential form. Assume all variables represent positive real numbers. 5.

x9

6.

( m) 5

5. ____________________________

4

6. ____________________________

Write in radical form. Assume all variables represent positive real numbers. 7. a1/ 6

7. ____________________________

8. (5m − n)8 / 5

8. ____________________________

Simplify each radical expression by changing the expression to exponential form. Write the answer in radical form.

9.

10.

5

610

9. ____________________________

18

x3

10. ____________________________

Simplify.

11.

45 x10 y 31

11. ____________________________

12.

x5 y 6 12 z

12. ____________________________

261

Chapter 7 Pretest Form B (cont.) 24 − 3 6 + 252

13.

14.

15.

(

6− 2

)(

3− 2

Name:

13. ____________________________

)

14. ____________________________

6

15. ____________________________

6 + 30

16. (5 + 7i) − (4 − 3i)

16. ____________________________

−16

17. ____________________________

18.

7x − 3 = 5

18. ____________________________

19.

x −8 = x − 2

19. ____________________________

17.

Solve.

20. A new bypass is being constructed to relieve congestion at an intersection. The road will run from 45 feet north of the perpendicular intersection to 60 feet east of it. How long is the new road?

262

20. ____________________________

Mini-Lecture 7.1 Roots and Radicals Learning Objectives: 1. 2. 3. 4. 5.

Find square roots. Find cube roots. Understand odd and even roots. Evaluate radicals using absolute value. Key Vocabulary: radical sign, radicand, radical expression, index, square root, principal square root, cube root, even root, odd root

Examples: 1. For the function f ( x) = 4 x + 9 , find each of the following: b) f (−2) c) a) f (10)

f (−5)

2. For the function f ( x) = 3 7 x − 8 , find each of the following: a) g (5) b) g (−5) c) g (19) 3. Indicate whether or not each radical expression is a real number. If the expression is a real number, find its value. a)

4

−625

b) − 4 625

c)

5

−243

d) − 5 −243

4. Use absolute value to evaluate. a)

52

b)

02

c)

(2.6) 2

d)

(−10) 2

e)

(2 x − 5) 2

f)

36 y 2

g)

64x 6

h)

400m 4

i)

n 2 − 20n + 100

Teaching Notes: • Remind students that the result of a square root is always nonnegative and that the result is only rational if the radicand is a perfect square. • Students may need extra practice to remember the difference between expressions like − 5 and −5 . • A quick review of factoring real numbers may be helpful as students begin simplifying radical expressions. • Encourage students to memorize as many perfect squares and perfect cubes as they can. • Stress that calculators only give approximate values for irrational numbers. The radical form is the exact value. • The textbook often assumes variables and radicands represent positive numbers. Remind students to pay attention to this detail in the instructions.

Answers: 1a) 7; 1b) 1; 1c) not a real number; 2a) 3; 2b) −4 ; 2c) 5; 3a) not a real number; 3b) real number, 5; 3c) real number, −3 ; 3d) real number, −3 ; 4a) 5; 4b) 0; 4c) 2.6; 4d) 10; 4e) 2x − 5 ; 4f) 6y or 6 y ; 4g) 8x3 or 8 x 3 ; 4h) 20m2 ; 4i) n − 10

263

Mini-Lecture 7.2 Rational Exponents Learning Objectives: 1. Change a radical expression to an exponential expression. 2. Simplify radical expressions. 3. Apply the rules of exponents to rational and negative exponents. 4. Factor expressions with rational exponents. 5. Key vocabulary: rational exponent, exponential expression, radical expression Examples: 1. Write each expression in exponential form.

a)

b)

13

3

6xy

5

c)

x2

d)

6

2m5 3n

2. Write each expression in radical form. b) (−64)1/ 3

a) 161/ 4

( 5x3 y )

1/ 5

c)

d) 6ab1/ 3

3. Write each expression in exponential form and then simplify.

(4 a)

8

x15

3

a)

b)

c)

5

m10 n15

4. Write each expression in radical form. a) x

3/8

b) (2ab)

2/7

c)

⎛ x⎞ ⎜ ⎟ ⎝7⎠

5. Simplify. a) 93 / 2

b)

6

43

10

c)

m5

d)

3/ 4

(9 a)

6

6. Evaluate. a) 81−3 / 4

b) 16−5 / 4

a) x1/ 4 ⋅ x −2 / 3

b)

⎛4⎞

−1/ 2

⎛ 125 ⎞

−1/ 3

c) ⎜ ⎟ d) ⎜ − ⎟ ⎝9⎠ ⎝ 8 ⎠ 7. Simplify each expression and write the answer without negative exponents.

(

3x 4 y −2

8. Simplify. 18

a)

(13n)6

b)

)

(

−1/ 2

1/ 4

⎛ 9 x −2 z1/ 3 ⎞ −2 / 3 ⎟ ⎝ z ⎠

d) 2.1x1/ 2 (1.6 x1/ 3 + x −1/ 4 )

c) ⎜

)

15

5

xy 3 z 4

c)

3 5

a

9. Factor r 3 / 8 + r −5 / 8 . Teaching Notes: • Demonstrate for students the convenience of using rational exponents to evaluate radical expressions on a calculator. For example, it is convenient to evaluate 161/ 4 for 4 16 .

Answers: 1a) 131/2 ; 1b) (6xy)1/3 ; 1c) x 2/5 ; 1d)

2m5 ; 2a) 3n

3a) x15/3 = x 5 ; 3b) a 8/4 = a 2 ; 3c) m10/5 n15/5 = m2 n3 ; 4a) 3

8

4

16 = 2 ; 2b)

x 3 or

3

(8 x)

3

3

−64 = −4 ; 2c)

; 4b)

7

(2ab)2 or

5

5x 3 y ; 2d) 6a 3 b ;

( 7 2ab )

2

;

⎛ x⎞ 3 1 1 2 or ⎜ 4 ⎟ ; 5a) 27; 5b) 2; 5c) m or m1/2 ; 5d) 3 a 2 or a 2/3 ; 6a) ; 6b) ; 6c) ; 6d) − ; 27 32 5 7 2 ⎝ ⎠ 31/2 z 1/4 1 y r +1 7a) 5/12 ; 7b) 1/2 2 ; 7c) ; 7d) 3.36x5/6 + 2.1x1/4 ; 8a) 3 13n ; 8b) x 3 y 9 z 12 ; 8c) 15 a ; 9) 5/8 1/2 x r 3 x x

4c)

4

⎛ x⎞ ⎜ ⎟ ⎝7 ⎠

264

Mini-Lecture 7.3 Simplifying Radicals Learning Objectives:

1. 2. 3. 4.

Understand perfect powers. Simplify radicals using the product rule for radicals. Simplify radicals using the quotient rule for radicals. Key vocabulary: product rule for radicals, quotient rule for radicals, perfect power, perfect square, perfect cube

Examples:

1. Simplify. Assume all variables represent non-negative real numbers. a)

400

b)

3

c)

1000

4

x 20

d)

x14

3

2. Simplify. Assume all variables and expressions in radicands are non-negative. a) e) i)

3

50

b)

63

c)

3

80

d)

4

162

x18

f)

y5

g)

4

a 35

h)

3

m16

x10 y13

j)

m10 n 21

k)

48a 7b10 c 5

l)

3 125 p14 q10

4

3. Simplify. Assume all variables and expressions in radicands are non-negative. 3

a)

63 7

b)

d)

49 x 4 144

e)

108 x

3

3

c)

4 x4

27 x5 y 64 x 2 y16

f)

3

a −2b11

3

a 4b −4

4

20 xy10 4 x13 y 2

Teaching Notes: • Warn students to be careful to pay attention to the type of root being taken. • Encourage students to memorize as many perfect powers as they can.

Answers: 1a) 20; 1b) 10; 1c) x 5 ; 1d) x 5 ; 2a) 5 2 ; 2b) 3 7 ; 2c) 2 3 10 ; 2d) 3 4 2 ; 2e) x6 ; 2f) y 2 y ; 2g) a 8 4 a 3 ; 2h) m 5 3 m ; 2i) x 5 y 6 y ; 2j) m 2 n 5 4 m 2 n ; 2k) 4a 3b 5 c 2 3ac ; 2l) 5p 4 q 3 3 p 2 q ; 3a) 3; 3b)

3x y2 4 5 3 b5 7x 2 ; 3c) 2 ; 3d) ; 3e) 5 ; 3f) 12 x x3 4y a 265

Mini-Lecture 7.4 Adding, Subtracting, and Multiplying Radicals Learning Objectives:

1. Add and subtract radicals. 2. Multiply radicals. 3. Key vocabulary: like radicals, unlike radicals Examples:

1. Simplify. Assume all variables represent non-negative real numbers. a) 8 + 5 7 − 7 + 6

b) 4 3 x − 6 + 5 3 x + 7 x

c)

d) 10 12 + 3 75

e) 3 63 − 4 28 + 175

f)

3

125 + 3 40 − 9 3 5

m6 − m 4 n + m 2 n

i)

3

a11b − 3 a 5b 7

32 + 12 − 18

g)

h)

18 + 72

2. Multiply and simplify. Assume all variables represent non-negative real numbers. a) c)

4

15 x 5 20 x 4

b)

9ab9 4 36a10b12

d)

e)

(

a + b )( a + b )

g)

( 3 m − 3 6n ) ( 3 m 2 + 3 9n 2 )

3

9 x 3 3x5 7 x ( 14 x + 28 )

f)

(2

10 − 5 )

h)

(5 +

2

7 )( 5 − 7 )

3. If f ( x) = 3 x and g ( x) = 3 x5 + 3 x 4 , find each of the following: a) ( f ⋅ g )( x ) b) ( f ⋅ g )( 3) 4. Simplify. Assume that variables may be any real number. a)

f ( x) = x − 2 x − 2, x ≥ 2

b) g ( x) = 5 x 2 − 30 x + 45

Teaching Notes: • A quick review of the previous discussion of collecting like terms may be helpful. • When adding or subtracting like radicals, students sometimes mistakenly multiply the like radicals.

Answers: 1a) 14 + 4 7 ; 1b) 9 3 x + 7x − 6; 1c) 9 2; 1d) 35 3; 1e) 12 7 ; 1f) 5 − 7 3 5; 1g) 2 3 + 2; 1h) m 3 ; 1i) (a 3 − ab 2 ) 3 a 2b ; 2a) 10x 4 3x ; 2b) 3x 2 ; 2c) 3a 2b5 4 4a 3b; 2d) 7x 2 + 14 x ; 2e) a + b a + ab + b b; 2f) 45 − 10 2; 2g) m + 3 9mn 2 − 3 6m 2 n − 3n 3 2 ; 2h) 18; 3a) x 2 + x 3 x 2 ; 3b) 9 + 3 3 9 ; 4a) f(x) = x − 2; 4b) g(x) = 5 x − 3

266

Mini-Lecture 7.5 Dividing Radicals Learning Objectives: 1. Rationalize denominators. 2. Rationalize a denominator using the conjugate. 3. Understand when a radical is simplified. 4. Use rationalizing the denominator in an addition problem. 5. Divide radical expressions with different indices 6. Key Vocabulary: rationalizing a denominator, conjugate Examples: 1. Simplify. Assume all variables represent non-negative real numbers.

1 11

a) e)

b)

5y 3

x 3 7 3 5

f)

2x2

c) g)

3

15 3x 3

d)

3m 2 4n

h)

4

56m5 3 n 405a 6b9 4c 3

2. Simplify. Assume all variables represent non-negative real numbers. 23 12 x+y b) c) a) 5− 2 7+ 3 x−y 3. Simplify each expression. Assume all variables represent non-negative real numbers. 150m9

a)

1 7

b)

4. Simplify: 2 5 +

1 10

c)

6 − 45 5

5. Simplify. Assume all variables and expressions in radicands are non-negative. a)

6

5

x x

b) 6

( a + b) 4

3

c)

( a + b)3

4

m4n2 m3 n

Teaching Notes: • Remind students that when they form the binomial conjugate, they only change the sign in the middle.

• Point out that, when we multiply by an expression such as

2+ 3 , we are really 2+ 3

multiplying by 1. Answers: 1a)

11 ; 11

1h)

3ab 2 4 20a 2 bc ; 2c

4)

5 ; 5

5a)

6

1b)

x 7 ; 21

1c)

5 3x ; x

1d)

2m 3 7m 2 n 2 ; n

2a) 5+ 2; 2b) 3 7 − 3 3; 2c)

x ; 5b)

20

a + b; 5c)

12

m7 n 5

267

1e)

5y 3 4x ; 2x

x + 2y x + y 2 ; x − y2

1f)

15 ; 5

3

1g)

3a) 5m4 6m; 3b)

6m 2 n 2 ; 2n 7 ; 7

3c)

10 ; 10

Mini-Lecture 7.6 Solving Radical Equations Learning Objectives:

1. 2. 3. 4. 5. 6.

Solve equations containing one radical. Solve equations containing two radicals. Solve equations containing two radical terms and a nonradical term. Solve applications using radical equations. Solve for a variable in a radicand. Key vocabulary: radical equation, isolating a radical, extraneous solutions

Examples:

1. Solve. a)

x =9

b)

x +5 −8 = 0

c)

3

x +9 = 7

d)

x +4=0

e)

2 x + 31 = x − 2

f)

x −4 x −5 = 0

2. Solve. a)

25 x 2 + 15 = 5 x 2 + x − 8

b) 4( x − 3)1/ 3 = (4 x + 48)1/ 3

3. Solve. a)

x −1 + x = 2

b)

5x −1 = 1 + 4x − 3

4. Solve each application. a) Find the length of the diagonal of a rectangle that is 10 feet long and 4 feet wide. b) Find the period of a pendulum if its length is 3 feet. The formula T = 2π

L , where L 32

is the length in feet, and T is the time in seconds. 5. Solve V =

2e for e. m

Teaching Notes:

• Remind students that solving an equation means to isolate the variable of interest and that for radical equations we are first trying to isolate the radical so we can square both sides. • Emphasize the importance of checking solutions in the original equation. • Be sure to explain clearly what an extraneous root is and why they might come about when solving a radical equation. Answers: 1a) 81; 1b) 59; 1c) −8; 1d) no real solution; 1e) 9; 1f) 25; 2a) 3b) 1, 13; 4a) 10.77 feet or 2 20 feet; 4b) ≈ 1.92 seconds; 5) e =

268

v2 m 2

43 ; 5

2b) 4; 3a)

25 ; 16

Mini-Lecture 7.7 Complex Numbers Learning Objectives:

1. 2. 3. 4. 5. 6.

Recognize a complex number. Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Find powers of i. Key vocabulary: imaginary numbers, the imaginary unit, complex numbers, conjugate of a complex number

Examples:

1. Write each complex number in the form a + bi . a) 11 − −81 d)

−72

b) 6 + −54

c) 12

e)

f) 3 + 5

99

2. Add or subtract. a) (−5 + 7i ) + (8 − 12i ) + 4

b)

3. Multiply. a) 4i (7 − 3i )

b)

4. Divide. 10 − 3i a) i

(6 −

−20 ) − ( −7 + −45 )

−36 ( −7 + 5 )

b)

c)

(3 −

−54 )( −6 + 4 )

2 + 5i 5 − 3i

5. Evaluate. a) i 46

b) i121

6. Let f ( x) = x 2 . Find each of the following: b) a) f (8i ) 7. Using the formula Z =

f (3 − 4i )

V , find the impedance, Z, when V = 2.1 + 0.4i and I = 0.5i . I

Teaching Notes:

• Remind students that the imaginary unit i is not a variable. Emphasize i = −1 . Answers: 1a) 11 − 9i; 1b) 6 + 3i 6 ; 1c) 12 + 0i; 1d) 0 + 6i 2; 1e) 3 11 + 0i; 1f) ( 3+ 5 ) + 0i; 2a) 7 − 5i; 2b) 13 − 5i 5 ; 3a) 12 + 28i; 3b) −6 7 + 30i; 3c) 30 − 9i 6 ; 4a) −3 − 10i; 4b)

−5 + 31i 34

or −

5 31 + i; 34 34

5a) −1; 5b) i; 6a) −64; 6b) −7 − 24i; 7) 0.8 − 4.2i

269

Name:

Additional Exercises 7.1

Date:

Evaluate the radical expression if it is a real number. If it is not real, indicate so. 1.

144

1. ____________________________

2.

25 81

2. ____________________________

3.

0.36

3. ____________________________

4.

2.25

4. ____________________________

−81

5. ____________________________

5.

4

6. Evaluate:

3

−64

6. ____________________________

Use absolute value to evaluate. 7.

( 29 )2

7. ____________________________

8.

( −0.14 )2

8. ____________________________

Write as an absolute value. 9.

10.

(5x

2

−y

)

2

9. ____________________________

( 4 x + 2 )2

10. ____________________________

Use absolute value to simplify. You may need to factor first. 11.

y16

11. ____________________________

12.

x 2 − 10 x + 25

12. ____________________________

13.

4a 2 − 28ab + 49b 2

13. ____________________________

270

Additional Exercises 7.1 (cont.)

Name:

Find the value of the function. Use your calculator to approximate irrational numbers to the nearest thousandth. 14. If f ( x ) = x + 1 , find f (15 ) .

14. ____________________________

15. If f ( x ) = 25 + 5 x , find f ( −3) .

15. ____________________________

3 16. If f ( x ) = 7 x 2 − 3 , find f ( 2 ) .

16. ____________________________

17. If f ( x ) = 4 4 x 2 − 3x + 9 , find f ( −1) .

17. ____________________________

3

18. Find the domain of

x −5

18. ____________________________

x +1

19. The velocity, v, of an object, in feet per second, after it has fallen a distance, h, in feet, can be found by the formula v = 64.4h . With what velocity will an object hit the ground if it falls from 30 feet? 20. Graph f ( x ) = x − 3 .

19. ____________________________

20.

y

x

271

Name:

Additional Exercises 7.2

Date:

Assume all variables represent positive real numbers. Write in exponential form. 1.

( x)

2.

5

9

4

1. ____________________________

y3

2. ____________________________

Write in radical form. 3.

(12b )

3. ____________________________

4.

1 3−1/ 2

4. ____________________________

5 2/3

Simplify each radical expression by changing the expression to exponential form. Write the answer in radical form, when appropriate. 5.

6.

7.

8.

(x) 3

4

( 5

2

3

5. ____________________________

x 28 3

a 2b

6. ____________________________

)

6

7. ____________________________

x7

8. ____________________________

Evaluate if possible. If the expression is not a real number, so state. 9.

( 25)−1/ 2 + ( 27 )−1/ 3

⎛ 121 ⎞ 10. ⎜ ⎟ ⎝ 4 ⎠

9. ____________________________

−1/ 2

10. ____________________________

Simplify. Write the answer in exponential form without negative exponents. 11. x5 / 4 ⋅ x5 / 2

11. ____________________________

12. x −1/ 2 ⋅ x −1/ 2

12. ____________________________

⎛ 16 x 3/ 5 ⎞ 13. ⎜ 1/ 3 ⎟ ⎝ 2x ⎠

2

13. ____________________________

272

Additional Exercises 7.2 (cont.) (

14. Multiply −4 y −3/ 8 5 y1/ 8 − y 2

15. Use a calculator to evaluate

) 5

Name: 14. ____________________________

209 . Round to the nearest hundredth.

15. ____________________________

Factor. Write the answer without negative exponents. 16. x1/ 2 + x3/ 2

16. ____________________________

17. x −5 + x −6

17. ____________________________

The formula used for carbon dating is P = P0 2−t / 5600 where P0 represents the original amount of carbon 14 ( C14 ) present and P represents the amount of C14 present after t years. 18. If 12 milligrams of C14 is present in a fossil now, how many milligrams will be present in 4000 years?

18. ____________________________

19. If 25 milligrams of C14 is present in a piece of bone now, how much will be present in 7000 years.

19. ____________________________

20. Find the domain of f ( x ) = ( x − 5 )

1/ 2

( x + 3)1/ 2 .

20. ____________________________

273

Name:

Additional Exercises 7.3

Date:

Simplify. Assume all variables represent non-negative, real numbers. 1.

176

1. ____________________________

2.

4

405

2. ____________________________

3.

3

192a 7 b3

3. ____________________________

4.

3

x5 y8

4. ____________________________

216x 2 y 5

5. ____________________________

1215a 6 b8

6. ____________________________

7.

100a100

7. ____________________________

8.

x4 y9

8. ____________________________

9.

1575

9. ____________________________

5. 6.

5

10.

108x 4 y 5

10. ____________________________

11.

3

375x14 y13

11. ____________________________

12.

3

49x8 y 4

12. ____________________________

13.

5

128a13b 23

13. ____________________________

14.

8x6 y 5

14. ____________________________

15.

32 2

15. ____________________________

16.

16 81

16. ____________________________

75 x 2

17.

17. ____________________________

3

18.

28 x 3 y 3 7 x3 y 5

18. ____________________________

19.

3

5 xy 27 x10

19. ____________________________

20.

3

3x6 y 9 48 x 2 y

20. ____________________________

274

Name:

Additional Exercises 7.4

Date:

Simplify. Assume all variables represent positive real numbers. 1. 5 4 9 − 8 4 9

1. ____________________________

2. 19 3 y − 8 3 y − 4 3 y

2. ____________________________

3. 6 x − 17 − 10 x + 24

3. ____________________________

4.

45 − 320

4. ____________________________

5.

63x 2 y + x 28 y

5. ____________________________

6.

3

640 − 3 80

6. ____________________________

7.

3

−8 x − 4 3 x 4 + 9 3 x + 3x 3 x

7. ____________________________

8.

3

64 x − 9 3 x 4 − 6 3 x + 6 x 3 x

8. ____________________________

12 x3 y 3 8 xy 2

9. ____________________________

9.

10.

11.

5

16a 7 b14 5 8a 6 b9

10. ____________________________

(

3

)

11. ____________________________

7 x4 y 2

2

12.

3 ( 75 + 27 )

13.

6y

(

12. ____________________________

12 y − y 3

14.

(

)(

15.

( −6

16.

(

17.

(

3

4−35

18.

(

3

x −2

2 +5

2 −4

x+ y

7 −5

)

)(

)

13. ____________________________

)

14. ____________________________

x +8 y

)

15. ____________________________

2

16. ____________________________

)(

)(

3

3

2 − 3 25

x2 − 7

)

17. ____________________________

)

18. ____________________________

275

Additional Exercises 7.4 (cont.) 19. Find the perimeter and area of the rectangle below. Write both in simplified radical form.

Name: 19. ____________________________

300 48

20. Find the perimeter and area of the triangle below. Write both in simplified radical form. 50

8

98

242

276

19. ____________________________

Name:

Additional Exercises 7.5

Date:

Simplify. Assume all variables represent non-negative, real numbers. h

1.

2.

1. ____________________________

6

5 6

2. ____________________________

3

3.

3 32

3. ____________________________

4.

6 x4 2 x5

4. ____________________________

5.

10

5. ____________________________

10

6.

18 x 3 3x 4

6. ____________________________

7.

2 x4 y5 8z

7. ____________________________

8.

6

r4 9s 20

8. ____________________________

9.

3

16 x 5 y 9 2 x6

9. ____________________________

10.

1 7

+

7 7

10. ____________________________

11.

1 + 72 8

11. ____________________________

12.

5 1 +3 + 45 2 5

12. ____________________________

13.

3

13. ____________________________

5− 6

277

Additional Exercises 7.5 (cont.) x

14.

15.

16.

17.

14. ____________________________

x+ 2 5− 3

15. ____________________________

5+ 3 6− 3

16. ____________________________

6+ 3 3

x

4

x

4

x10 y 6

6

x2 y 4

18.

Name:

17. ____________________________

18. ____________________________

The radius, r, of a sphere with volume v can be found by the formula r =

3

3v . 4π

19. At a state fair, a spherical decoration is to have a volume of 1436 cubic feet. Find the approximate radius of the decoration.

19. ____________________________

20. Find the radius of a ball for a circus act, if the ball must have a volume of 113,040 cubic inches.

20. ____________________________

278

Name:

Additional Exercises 7.6

Date:

Solve and check your solution(s). If the equation has no real solution, so state.

x−5 = 2

1. ____________________________

x − 11 + 2 = 9

2. ____________________________

x−9 = 6

3. ____________________________

4.

x 2 + 5 x − 10 = x

4. ____________________________

5.

x 2 − x − 30 = x + 5

5. ____________________________

6.

4− x +4 = x

6. ____________________________

7.

m+9 +3 = m

7. ____________________________

8. (4a − 9)1/ 4 = (a − 3)1/ 4

8. ____________________________

9.

x + 2 = 4 + 9x

9. ____________________________

10.

x − 11 − x = −1

10. ____________________________

11.

x+4 = x −4

11. ____________________________

1.

3

2. 3.

3

12. 5 + x = 25 + 10 x

12. ____________________________

Given f ( x ) and g ( x ) , find all real values of x where f ( x ) = g ( x ) . 13.

f ( x) = x + 5 , g ( x) = 2x − 3

13. ____________________________

14.

f ( x ) = 3 x 2 − 3 x + 5 , g ( x ) = 3 x 2 + 7 x − 15

14. ____________________________

15.

f ( x ) = ( 20 x − 11) 2 , g ( x ) = 3 ( 4 x − 3) 2

15. ____________________________

1

1

Solve each formula for the variable indicated. 16. u =

HR for H a

16. ____________________________

17. r =

qX for y y

17. ____________________________

The formula for the period of a pendulum on the moon is T = where T is the period in seconds and A is the length in feet.

π 3A , 2

18. Find the period of a pendulum on the moon whose length is 12 feet.

18. ____________________________

19. Solve the formula for the period of a pendulum on the moon for A .

19. ____________________________

20. Find the length of a pendulum that has a period of 5 seconds on the moon.

20. ____________________________

279

Name:

Additional Exercises 7.7

Date:

Write each expression as a complex number in the form a + bi . 1. −7 + −100

1. ____________________________

2. 3i − −64

2. ____________________________

Perform the operations as indicated. 3.

( −6 − 9i ) − (8 + 5i )

3. ____________________________

4.

( −13 + 7i ) − ( −4 − 11i )

4. ____________________________

5.

−108 + −75

5. ____________________________

6.

−98 + −32

6. ____________________________

7.

(14 +

) (

−63 −

49 − −28

)

7. ____________________________

8. −4 ( 3 − 5i )

8. ____________________________

9. 3i (11 − 4i )

9. ____________________________

10.

( 5 + i )( −5 − 6i )

10. ____________________________

11.

(8 + i )( −4 + 9i )

11. ____________________________

12.

(8 −

12. ____________________________

13.

7 4i

13. ____________________________

14.

3 + 8i 2i

14. ____________________________

15.

1 − 4i 9+i

15. ____________________________

16.

)

⎛1 ⎞ −18 ⎜ + −2 ⎟ 4 ⎝ ⎠

2

16. ____________________________

7 − −343

Indicate whether the value is i, –1, –i, or 1. 17. i 203

17. ____________________________

18. i112

18. ____________________________

19. If f ( x ) = x 2 , find f ( 2 − i ) .

19. ____________________________

20. If f ( x ) = x 2 − 5 x , find f ( 3 + i ) .

20. ____________________________

280

Name:

Chapter 7 Test Form A 1. Evaluate: − 3 − 125

1. ____________________________

25 144

2. Evaluate:

Date:

2. ____________________________

4 x 2 − 12 x + 9

3. Write as an absolute value:

(

4. Write in radical form: 7b 2 c

5. Write in exponential form:

)

3. ____________________________

35

4. ____________________________

x3 y 2

5

5. ____________________________

For questions 6 – 17, assume all variables represent positive real numbers. 6. Simplify:

4

7. Simplify:

x12

6. ____________________________

75

7. ____________________________

8. Simplify: − 20x6 y 7 z12

9. Simplify:

4

8. ____________________________

20 x 4 81x − 8

9. ____________________________

8 − 12

10. Simplify:

10. ____________________________

11. Simplify:

(

x+y

)(

x−y

)

11. ____________________________

12. Simplify:

4

3x9 y12

4

54 x 4 y 7

12. ____________________________

13. Simplify:

x

13. ____________________________

13

14. Rationalize the denominator:

15. Rationalize the denominator:

5

14. ____________________________

2 +1

c − 2d

15. ____________________________

c− d

281

Chapter 7 Test Form A 16. Simplify:

17. Divide:

2 50

− 3 50 −

(cont.)

1

Name:

16. ____________________________

8

12 − − 12

17. ____________________________

3 + −5

18. Solve and check solution(s):

z2 + 3 = z + 1

18. ____________________________

19. Solve and check solution(s):

x +1 = 2 − x

19. ____________________________

20. Write the complex number 21 − − 36 in the form a + bi.

282

20. ____________________________

Name:

Chapter 7 Test Form B 1. Evaluate :

3

1 27

1. ____________________________

2. Use absolute value to evaluate

3. Write in radical form:

(7x

2

( 5 x − 8 )2 .

+ 2 y3

)

2. ____________________________

−1 6

3. ____________________________

y6

4. Simplify:

Date:

4. ____________________________

5. Simplify and write the answer in exponential form without

5. ____________________________

14 3 12

⎛ 81z y ⎞ negative exponents: ⎜ ⎟ 14 ⎝ 9z ⎠

(

6. Multiply: − 9z 3 2 z 3 2 − z − 3 2

7. Simplify:

4

)

6. ____________________________

80

7. ____________________________

For questions 8 – 9, assume all variables are positive. Simplify the given expressions. 8.

9.

4

48x11 y 21

8. ____________________________

150a10b11

9. ____________________________

2ab 2

Simplify each expression in questions 10 – 12. 10. 3 5 + 500 − 80

11.

12.

(3

10. ____________________________

a − 7 b )( 3 a + 7 b )

11. ____________________________

3 ( 75 + 15 )

13. Simplify:

12. ____________________________

20 y 4 z 3 3 xy − 2

13. ____________________________

283

Chapter 7 Test Form B

(cont.)

Name:

Rationalize the denominator in problems 14 – 15. 14.

15.

5

14. ____________________________

6+ 5

2

15. ____________________________

x+2 −3

16. Simplify: 2

8 100 −4 3 6

16. ____________________________

Solve and check your solution(s) in questions 17 – 18. 17.

x + 2x = 1

17. ____________________________

18.

y +1 =

18. ____________________________

19. Add:

(

20. Divide:

y+5−2

) (

20 − −12 + 2 5 + − 75

)

19. ____________________________

10 + − 3

20. ____________________________

5 − − 20

284

Name:

Chapter 7 Test Form C 1. Use absolute value to evaluate

( −6 ) 2

.

1. ____________________________

x 2 − 10 x + 25

2. Write as an absolute value.

Date:

2. ____________________________

1/ 3

⎛ 27 x3 ⎞ 3. Simplify ⎜ 9 ⎟ ⎝ −y ⎠

3. ____________________________

4. Graph f ( x ) = x − 2

4.

y

x

5. Write in exponential form:

3

x2 y

5. ____________________________

Simplify. Assume that all variables represent positive real numbers. 6.

24a9 b6

6. ____________________________

7.

14 x 7 xy 2

7. ____________________________

60 x 4

8.

7 3x

9.

10.

8. ____________________________

12 x

9. ____________________________

−2

10. ____________________________

1− 2

8 + 2 32

11.

11. ____________________________

12. 7 64 x − 2 25 x − 4 36 x 13.

( 6 + 2 )( 2 − 2 )

14.

3

3

15.

12. ____________________________ 13. ____________________________

xy 2

14. ____________________________

x4

15. ____________________________

x

285

Chapter 7 Test Form C

(cont.)

Name:

Solve. 16.

17.

3

n−2 = 4

16. ____________________________

13 − x + 1 = x

17. ____________________________

18. Multiply ( 2 + 5i ) ( 4 − 2i )

18. ____________________________

2 + 3i 1 − 2i

19. ____________________________

19. Divide

20. Evaluate x 2 − x + 1 for x = 3 − 2i .

20. ____________________________

286

Name:

Chapter 7 Test Form D 1. Use absolute value to evaluate

( −9 ) 2

.

1. ____________________________

4 x2 − 4 x + 1

2. Write as an absolute value.

Date:

2. ____________________________

3. Simplify x1/ 2 ⋅ x1/ 4 .

3. ____________________________

4. Graph f ( x ) = x + 2

4.

y

x

5. Write in exponential form:

6

xy 5

5. ____________________________

Simplify. Assume that all variables represent positive real numbers. 6.

45x 2 y 3 z 5

6. ____________________________

7.

2 x 18 y 2

7. ____________________________

8.

42a 3b5 14a 2 b

8. ____________________________

9.

6 5

9. ____________________________

10.

3

10. ____________________________

2− 3

81 + 6 3 3 − 3 24

11. ____________________________

12. 2a 27ab5 + 3b2 3a3b

12. ____________________________

11.

13.

3

(

)(

5 −3

)

13. ____________________________

72x6 y 4 z

14.

15.

5 −6

3

x5

4

x3

14. ____________________________

15. ____________________________

287

Chapter 7 Test Form D

(cont.)

Name:

Solve. 16.

7x − 3 = 2

16. ____________________________

17.

2x −1 = x − 2

17. ____________________________

18. Multiply ( 4 + 3i ) ( 6 + i )

18. ____________________________

19. Divide

−4 − 7 i 6i

19. ____________________________

20. Evaluate x 2 + 5 x − 2 for x = 2 − 3i .

20. ____________________________

288

Name:

Chapter 7 Test Form E

Date:

2

1. Use absolute value to evaluate

1. ____________________________

x2 + 2 x + 1

2. Write as an absolute value. ⎛ x −1/ 3 ⎞ 3. Simplify ⎜ 2 ⎟ ⎝ y ⎠

⎛ 1⎞ ⎜− ⎟ . ⎝ 8⎠

2. ____________________________

3

3. ____________________________

4. Graph f ( x ) = − x

4.

y

x

5. Write in exponential form:

10

x9 y 7 z 3

5. ____________________________

Simplify. Assume that all variables represent positive real numbers.

−8x5 y 9

6. ____________________________

7.

27 x3 3xy 2

7. ____________________________

8.

75a 4 b 7 3ab3

8. ____________________________

7 9

9. ____________________________

6.

9.

10.

11.

3

3

4

10. ____________________________

3+ 2

3 2 1 + 2 2

11. ____________________________

12. 3 8 x 2 + 2 18 x 2 − 4 x 2 13.

(

3

14.

15.

7 −2

)(

7 −8

12. ____________________________

)

13. ____________________________

x2 y 4

14. ____________________________

x 4

15. ____________________________

x

289

Chapter 7 Test Form E

(cont.)

Name:

Solve.

4x − 3 = 5

16. ____________________________

2 x + 32 − 4 = x

17. ____________________________

18. Multiply ( 2 + 3i )( 4 + 5i )

18. ____________________________

16.

17.

3

19. Divide

4 − 5i 2i

19. ____________________________

20. Evaluate x 2 − 3 for x = 2 − 5i .

20. ____________________________

290

Name:

Chapter 7 Test Form F

Date:

2

⎛ 4⎞ ⎜− ⎟ . ⎝ 9⎠

1. ____________________________

x 2 − 20 x + 100

2. ____________________________

1. Use absolute value to evaluate

2. Write as an absolute value. 3. Simplify x 2 / 5 ⋅ x 2

3. ____________________________

4. Graph f ( x ) = x + 2

4.

y

x

5. Write in radical form: ( x 3 y 2 )

1/ 5

5. ____________________________

Simplify. Assume that all variables represent positive real numbers. 6.

50x 2 y 6

7.

3

8.

9.

(

6. ____________________________

27 − 3

)

7. ____________________________

48 xy 7 6 y4

3

8. ____________________________

5 3

9. ____________________________

a 2

10.

10. ____________________________

5+2

300 − 12 + 3

11.

11. ____________________________

12. 5 9 x − 3 4 x + 6 16 x 13.

(

6+4

14.

3

x4 y3

15.

)(

6 −4

12. ____________________________

)

13. ____________________________

14. ____________________________

x 3

15. ____________________________

x

291

Chapter 7 Test Form F

(cont.)

Name:

Solve.

3x + 2 = 2

16. ____________________________

2 x + 14 = x + 3

17. ____________________________

18. Multiply ( 3 + 2i )( 5 + 4i )

18. ____________________________

16.

17.

3

19. Divide

−5i 2 − 4i

19. ____________________________

20. Evaluate x 2 + x + 1 for x = 2 − i .

20. ____________________________

292

Name:

Chapter 7 Test Form G 1. Use absolute value to evaluate. (a) –16

x+4

(a)

( −16 )2

(b) 16

2. Write as an absolute value.

Date:

(c) –4

(d) 4

x 2 − 8 x + 16 (b) x − 4

(c)

x 2 − 8 x + 16

(d) x 2 + 8 x + 16

1 x y

(c)

1 xy

(d)

5

(c) x 6 x

3

⎛ x −1/ 3 ⎞ 3. Simplify ⎜ 1/ 3 ⎟ . ⎝ y ⎠

y3 x3

(a)

(b)

3 3

y x

4. Write x 6 / 5 as a simplified radical. 6

(a)

x5

(b)

x6

(d) x 5 x

5. Graph f ( x ) = x − 2 y

(a)

y

(b)

y

(c)

y

(d)

4

4

4

4

2

2

2

2

—4 —2 —2

2

4

x

—4 —2 —2

—4

2

4

x

—4 —2 —2

—4

4

x

—4 —2 —2

—4

—4

Simplify. Assume that all variables represent positive real numbers.

8x3 y 5

6.

(a) 4 xy 2 2 xy 7.

3

(b) 4 3 2a

(c) 4a 3 2

(d) 2a 3 4

(b) 5y

(c) 5 y

(d)

5y

(d)

ab 7 b

8x

7a 2 b

(a) a 7b

10.

(d) 2 xy 3 4 xy 2

40 xy 3

(a) y 5 y

9.

(c) 4 x 2 y 4 2 xy

4a 2 3 8a

(a) 2 3 4a

8.

(b) 2 xy 2 2 xy

(b)

a 7b b

(c)

ab 7

7 2− 3

(a) 14 − 7 3

(b) 14 + 7 3

(c) −14 − 7 3

293

(d) −14 + 7 3

2

4

x

Chapter 7 Test Form G 11.

3

(cont.)

Name:

16 − 2 3 2

(a) 0

(b) 2 3 2

(c) 4 3 2

(d) 8 3 2

(b) 101 5x

(c) −17 5x

(d) 51 5x

(b) 4 + 2 3

(c) 4 + 3

(d) 4 + 6

(b) 4 3 4

(c)

3

2

(d) 4 3 2

(c)

15

x7

(d)

12. 4 20 x + 5 45 x − 10 80 x (a) 33 5x 13.

(

)

3 +1

2

(a) 4 14.

8 3

2 4

(a) 2 3 4 5

15.

x4

3

(a)

x 5

x3

(b)

3

x

15

x3

x−2 −6 = 0

16. Solve. (a) 4

(b) 8

(c) 38

(d) no real solution

(b) 2

(c) 9

(d) no real solution

(b) –1

(c) −1 + 6i

(d) 17 + 6i

x+7 = x−5

17. Solve. (a) 2, 9

18. Multiply ( 2 + 3i )( 4 − 3i ) (a) 8 − 9i 19. Divide (a)

4 5 − 2i

20 − 8i 3

(b)

20 − 8i 29

(c)

20 + 8i 3

(d)

20 + 8i 29

20. Evaluate x 2 − 2 x + 3 for x = 1 − i . (a) 3

(b) 0

(c) 1

294

(d) –1

Name:

Chapter 7 Test Form H 1. Use absolute value to evaluate (a) –6

(a)

( −36 )2

(b) 6

2. Write as an absolute value

x−4

Date:

(c) –36

(d) 36

x 2 − 8 x + 16 (b) x + 4

(c)

(b) x1/10

(c) x 2 / 5

x 2 − 8 x + 16

(d) x 2 + 8 x + 16

3. Simplify x1/ 5 ⋅ x1/ 2 . (a) x 7 /10

(d) x5 / 2

4. Write x 7 / 5 in simplified radical form. (a) x 7 x 2

(b) x 5 x 2

(c)

7

x5

(d)

y

(d)

5

x7

5. Graph f ( x ) = x + 1 y

(a)

y

(b)

(c)

y

4

4

4

4

2

2

2

2

2

4

6

8

x

2

4

6

8

x

2

4

6

8

x

2

—2

—2

—2

—2

—4

—4

—4

—4

(b) 2 xy 8 y 2

(c) 8 xy 4 xy

(d) 4 xy 2 y

(b) 2 x 2 3 2

(c) 4 x 3 2 x

(d) 2 x 3 2 x 2

(b) 5y x

(c) 5x y

(d)

Simplify. Assume that all variables represent positive real numbers.

32x 2 y 3

6.

(a) 8xy 2 xy 7.

3

x 2 3 16 x 4

(a) 2 x 3 2 x

50 x3 y 2 2 xy

8.

(a) 5 xy

9.

3

3 5 3

(a)

10.

5xy

45 5

3

3

(b)

15 5

(c)

75 5

(b)

3−2

(c) − 3 + 2

(d)

33 5 5

−1

3−2

(a)

3+2

295

(d) − 3 − 2

6

8

x

Chapter 7 Test Form H

(cont.)

Name:

11. 3 27 + 5 12 (a) 16 6

(b) 47 3

(c) 19 2

(d) 19 3

(b) 7 3n

(c) −2 3n

(d) −11 3n

(b) −2 + 4 3

(c) 2 − 4 3

(d) 2 + 4 3

(c)

4

x2 y

(d) x y

x

(c)

12

x5

(d)

59 3

(c)

12. 5 27n − 12n − 6 3n (a) 35 3n 13.

(

)(

3 +1

3 −5

)

(a) −2 − 4 3 x2 y

14. (a)

15.

4

x3

3

x2

(a)

3

x2 y

(b) x 4 y

3

x4

(b)

(a) 21

(b)

(a) 6, 3

(b) 6

(

)(

18. Multiply 5 − −2 3 + −2 (a) 17 + 2i 2

i 5

x5

1 3

(d) no real solution

(c) 3

(d) no real solution

(c) 17 − 2i 2

(d) 13 − 2i 2

x−2 = x−4

17. Solve.

(a)

4

3x − 5 = 8

16. Solve.

19. Divide

12

) (b) 13 + 2i 2

4 5i

5i 4

(c) −

(b) –9 – 12i

(c) –9

(b)

4i 5

(d)

4i 5

20. Evaluate x 2 − 4 for x = 2 − 3i . (a) 9

296

(d) 9 – 12i