124b End of Semester Practice Problems Name____________________________

Simplify the radical.

1) 3 127

1)

2)4

256 2)

3)4

-16 3)

4)3

83 4)

5)5

(-3)5 5)

1

6)3

(x + 2)3 6)

Evaluate the expression, if possible.7) 641/2 7)

8) 2764

-1/38)

Rewrite the expression with a positive rational exponent. Simplify, if possible.

9) y3/4

y1/4 9)

10) 2-2/7 · 23/7 10)

2

Simplify by factoring out the given factor.11) 8k-3 + k-7; k-7 11)

Use the product rule to multiply. Assume all variables represent positive real numbers.12) 11 · 11 12)

13)3

2 ·3

4 13)

Use the product rule to simplify the expression. Assume that the variables can be any real number.

14)3

-125 14)

15) x24 15)

16) 112 x 2y 16)

3

Use the quotient rule to divide and simplify.

17) 825

17)

18) 966

18)

19) 63x117x

19)

Add or subtract. Assume all variables represent positive real numbers.20) 3 5 + 8 125 20)

21) 9 + 720 + 121 + 80 21)

4

22) 43

3 + 103

3 22)

Rationalize the denominator. Assume that all variables represent positive real numbers.

23) 413

23)

24) -7

2 524)

25) 511

25)

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.

26) 62 - 9

26)

5

27) 75 + 2

27)

28) 23 + 5

28)

29) zz + m

29)

Evaluate the radical function at the indicated value.30) f(x) = - 2x - 6

f(21)30)

Find the domain of the given function.31) f(x) = 8 - 7x 31)

6

32) f(x) =3

5 - 4x 32)

33) f(x) =4

10 x - 2 33)

Determine the domain and range of the function. Then graph it.34) f(x) = x + 5 34)

Use the square root property to solve the equation.35) x2 = 169 35)

7

36) x2 = 15 36)

37) 5x2 = 121 37)

38) (x - 7)2 = 36 38)

39) (x + 8)2 = 20 39)

Complete the square for the binomial. Then factor the resulting perfect square trinomial.40) x2 - 16x 40)

41) x2 - 7x 41)

8

Solve the equation by completing the square.42) x2 - 2x - 8 = 0 42)

43) x2 + 3x - 9 = 0 43)

44) x2 + x + 9 = 0 44)

45) 4x2 + 1 = 3x 45)

Use the quadratic formula to solve the equation.46) x2 + 6x + 9 = 0 46)

47) 2x2 - 5x - 7 = 0 47)

9

48) 2x2 = -8x - 7 48)

49) 16x2 + 1 = 7x 49)

Determine the discriminant of the quadratic equation. Use the value of the discriminant to determine whether thequadratic equation has two rational solutions, two irrational solutions, one repeated real solution, or two complexsolutions that are not real.

50) x2 + 12x + 36 = 0 50)

51) x2 - 3x + 3 = 0 51)

52) 36x2 - 12x = 0 52)

53) 4 + 5x2 = 5x 53)

10

Solve the problem.54) The hypotenuse of an isosceles right triangle is 2 feet longer than either of its legs. Find the

exact length of each side.54)

55) The area of a rectangular wall in a classroom is 260 square feet. Its length is 4 feet shorterthan three times its width. Find the length and width of the wall of the classroom.

55)

Solve the equation.56) x4 - 40x2 + 144 = 0 56)

57) x - 12 x + 35 = 0 57)

58) (2x - 4)2 + 2(2x - 4) - 3 = 0 58)

11

For the given functions f and g, find the indicated value.59) f(x) = x2 + 6x, g(x) = x + 2

(f g)(4)59)

60) f(x) = 4x + 6, g(x) = 4x2 + 3(f g)(0)

60)

61) f(x) = 3x3, g(x) = -x2 + 4(g g)(3)

61)

Indicate whether the function is one-to-one.62) Town Number of Commissioners

Winthrop -------------> 11Lincolnshire -----------> 16Baker -----------------> 16Middleville ------------> 9

62)

Find the inverse of the function.63) {(7, 1), (5, 2), (3, 3), (1, 4)} 63)

12

Indicate whether the graph represents a one-to-one function.64) 64)

Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph ofy = x is also given.

65) 65)

Find the inverse of the one-to-one function.66) f(x) = (x + 6)3 66)

13

67) f(x) = 2x 67)

68) f(x) =5x 68)

69) f(x) =3

x - 5 69)

Determine whether the functions f and g are inverses of each other.

70) f(x) =8

x - 6; g(x) =

8x

+ 6 70)

Approximate the value using a calculator. Express answer rounded to three decimal places.71) 2e 71)

14

Graph the exponential function.

72) f(x) = 3x + 1 72)

73) f(x) = ( 12

)x

73)

15

74) y = -3x 74)

Solve the equation.

75) 3x =19 75)

76) 5-x =125 76)

77) 2x =116 77)

16

78) 13

x- 9 = 0 78)

79) 3(x2 - 12) = 81x 79)

Solve the problem.80) The expected future population of a small town, which currently has 7900 residents, can be

approximated by the formulay = 7900(1.6)-0.3t

where t is the number of years in the future. Find the expected population of the town 40years in the future.

80)

Change the exponential expression to an equivalent expression involving a logarithm.

81) 4-3 =164 81)

82) 8x = 64 82)

17

83) ex = 6 83)

Change the logarithmic expression to an equivalent expression involving an exponent.

84) log 3127

= -3 84)

85) log 7 49 = x 85)

86) logb 49 =23 86)

Evaluate the expression without using a calculator.87) log10100 87)

88) log121 88)

18

89) log12 12 89)

90) log93 90)

Find the domain of the logarithmic function.91) f(x) = log(x - 4) 91)

Use a calculator to approximate the logarithm to four decimal places.92) log 2.04 92)

Use a calculator to approximate the natural logarithm to four decimal places.93) ln 5 93)

Solve.

94) log 4116

= x 94)

19

95) log 2 1 = x 95)

96) log4164

= y 96)

97) log 6 (x - 4) = 2 97)

98) ln(x) = 4 98)

Solve the problem.99) The Richter Scale measures the magnitude M of an earthquake. An earthquake whose

seismographic reading measures x millimeters 100 kilometers from the epicenter has

magnitude M given by M(x) = log x10-3

. Give the magnitude of an earthquake that

resulted in a seismographic reading of 69,003 millimeters 100 kilometers from its epicenter.

99)

20

Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.100) ln e2 2 100)

101) log2 28 - log2 14 101)

102) eln 6 102)

103) 10log 18 - log 3 103)

Write as the sum and/or difference of logarithms. Express powers as factors.

104) logbxy3

z5 104)

105) log 1619 r

s105)

21

106) log 5x

25106)

Write the expression as a logarithm of a single expression. Assume that variables represent positive numbers.107) log 2 7 + log 2 12 107)

108) log 8 12 + log 8 4 - log 8 16 108)

109) 4 log b m - log b n 109)

110) 12

(log7 (r - 1) - log7 r) 110)

Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round to four decimal places.111) log2071 111)

22

112) log 1/2 5 112)

Solve the equation.113) log3 (x + 5) + log3 (x - 1) = 3 113)

114) 12

log7 x = log7 8 114)

115) log3 (x - 5) + log3 (x - 11) = 3 115)

116) log x + log (x - 1) = log 42 116)

Solve the equation. Give an exact solution.117) 3x + 8 = 2 117)

23

118) e(x + 8) = 5 118)

119) 47x = 3.6 119)

Solve the equation. Give an approximate solution to four decimal places.120) 2x + 6 = 3 120)

121) e(x + 6) = 4 121)

Solve the problem.122) The value V of a car that is t years old can be modeled by V(t) = 19,655(0.83)t. According to

the model, when will the car be worth $6000?122)

123) The formula P = 112e0.028t models the population of a particular city, in thousands, t yearsafter 1998. When will the population of the city reach 125 thousand?

123)

24

Find the distance d(P1, P2) between the points P1and P2.124) P1 = (-4 7, 3); P2 = (-3 7, 6) 124)

Find the midpoint of the line segment formed by joining the points P1 and P2.125) P1 = (3, 1); P2 = (-5, 6) 125)

Find the center and radius of the circle whose graph is shown. Write the standard form of the equation of the circle.126) 126)

Write the standard form of the equation of the circle whose radius is r and whose center is (h, k).127) r =8; (h, k) = (0, 7) 127)

25

Find the standard form of the equation of the circle.128) Center at (5, -6) and containing the point (3, 2). 128)

Find the center (h, k) and the radius r of the circle.129) x2 + y2 - 4x - 12y + 25 = 0 129)

Write the first four terms of the sequence.130) {4n - 2} 130)

131) 35

n131)

The given pattern continues. Write down the nth term of the sequence suggested by the pattern.132) 0, 2, 6, 12, 20, ... 132)

26

133) 11

, 14

, 19

, 116

, 125

, ... 133)

Write out the sum. Do not evaluate.134)

6

k = 1(5k - 3)

134)

Find the indicated sum.

135)5

i = 2(4i - 4) 135)

136)4

i = 1

13i

136)

137)4

i = 12i 137)

27

138)4

i = 1-

13

i138)

Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.139) 3 + 12 + 27 + . . . + 75 139)

140) 14

+25

+12

+ . . . +1013 140)

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index ofsummation.

141) 25 + 35 + 45 + . . . + 95 141)

142) 3 +72

+ 4 +92

+ . . . +152 142)

28

Determine whether the sequence is arithmetic. If the sequence is arithmetic, determine the first term a and commondifference d.

143) 4, 12, 36, 108, 972, . . . 143)

144) 2, -1, -4, -7, -10, . . . 144)

An arithmetic sequence is given. Find the common difference and write out the first four terms.145) {7n + 6} 145)

Find a formula for the nth term of the arithmetic sequence whose first term a and common difference d are given.146) a = 20, d = -7 146)

Write a formula for the nth term of the arithmetic sequence. Use the formula to find the 20th term of the sequence.147) 10, 1 , -8, -17, . . . 147)

29

148) a =35

, d = -35 148)

Evaluate the given binomial coefficient.

149) 105 149)

Expand the expression using the Binomial Theorem.150) (x - 2)5 150)

151) (x2 + 5y)4 151)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the term indicated in the expansion.152) (x2 + y4)9; 3rd term 152)

A) 25920x14y8 B) 36x9y6 C) 36x14y8 D) 25920x9y6

30