Math 106: Worksheet 15

Simplify.

1.(a3b4c5)4

(a2b3c4)2

a8b10c12

2. (−2x3)0

1

3. (−2x2y3)4(4xy3)

64x9y15

4. (2x− y)0

1

5. 2 · 50 − 5

−3

6.6x−5

5y−1

6y

5x5

7.

(x−4

9−1

)2

81

x8

8.

(6ab−2

3a2b−4

)3

8b5a3

9.a−2c3

2b−5

b5c3

2a2

10.

(c−4c−2

c−6c8

)−4

c32

1

11. (z − 7)(z + 7)

z2 − 49

12. (a− 4)(a+ 4)

a2 − 16

13. (w + 8)2

w2 + 8w + 64

14. (r + 10)2

r2 + 20r + 100

15. (j − 5)2

j2 − 10j + 25

16. (h− 4)2

h2 − 8h+ 16

17. (2b2 − 1)(2b2 + 1)

4b4 − 1

18. (1− 3c)2

1− 6c+ 9c2

Factor.19. a2 − 4

(a− 2)(a+ 2)

20. b2 − 9

(b− 3)(b+ 3)

21. c2 − 36

(c− 6)(c+ 6)

2

22. 2d3 − 98d

2d(d− 7)(d+ 7)

23. j3 − jk2

j(j − k)(j − k)

24. 32x2y − 2y3

2y(4x− y)(4x+ y)

Solve.

25.z

z − 2− 1

z + 5=

7

z2 + 3z − 10

z = 1

26.3

p− p+ 3

3p=

2p− 1

2p− 5

6

p = 5

27.4

c− 2− 1

2− c=

25

c+ 6

c = 4

28.−3

w + 2=

w

w + 2

w = −3

Simplify.

29.

1

x+ 1+ 1

3

x+ 1+ 3

1

3

30.

1− 3

y + 1

3 +1

y + 1y − 2

3y + 4

31.x− x− 6

x− 1

x− x+ 15

x− 1

3

x2 − 2x+ 6

(x− 5)(x+ 3)

32.

1

z − 3− 4

z3

z − 3+

1

z12− 3z

4z − 3

33.

2

r − 1− 3

r + 14

r + 1+

5

r − 15− r

9r + 1

Find the domain of each expression.

34.√2x− 5

[5

2,∞)

35.√3x+ 12

[−4,∞)

36.√x+ 5

[−5,∞)

37.√x

[0,∞)

Simplify the complex fractions.

38.

3

2m+

4

3m1

4m− 5

9m102

11

39.

1

x+ 1+ 1

3

x+ 1+ 3

1

3

4

40.

1− 3

y + 1

3 +1

y + 1y − 2

3y + 4

50.z − z − 6

z − 1

z − z + 15

z − 1z2 − 2z + 6

(z − 5)(z + 3)

51.

1

q − 3− 4

q3

q − 3+

1

q12− 3q

4q − 3

52.

2

p− 1− 3

p+ 14

p+ 1+

5

p− 15− p

9p+ 1

Solve each equation.

53.14

a2 − 1+

1

a− 1=

3

a+ 1

9

54. b− 3b

2− b=

6

b− 2

−3

55.c

6=

3c

11

0

56.3

p− p+ 3

3p=

2p− 1

2p− 5

6

5

57.h+ 17

h2 − 1− 1

h+ 1=

h− 2

h− 1

5

−4, 5

58.3

k − 2− 5

k + 3=

1

k2 + k − 6

9

59.5

2m+ 6− 1

m+ 1=

1

m+ 3

3

Simplify60.

√8w3y3

2wy√2wy

61. 3√

48x3y8z7

2xy2z2 3√

6y2z

62.3

√−27y36

1000

−3y12

10

63.

√9a2

49b2

3a

7b2

64.

2a12

b13

6

64a3

b2

65. (km12 )3(k3m5)

12

k92m4

66. (m14n

12 )2(m2n3)

12

m32n

52

67.

(a−1m b

−1n

)−mn

6

anbm

68.(a−m2 b

−n3

)−6

a3mb2n

69.

(a−3mb−6n

a9m

)−13

anbm

70.

a−3m b

6n

a−6m b

9n

−13

b1n

a1m

Simplify.

71.√2t5 ·√10t4

2t4√5t

72. (3√2−√3)(2√2 + 3

√3)

3 + 7√6

73. (5√j − 3)2

25j − 30√j + 9

74. 10√m−

√16m

6√m

75. 5√z2x− 7

√z2x+ 6

√z2x

4z√x

Simplify. Rationalize denominators if needed.

76.1 +√2√

3− 1√2 +√3 +√6 + 1√

4

7

77.6

3−√18

−2− 2√2

78.5√

7−√5

5√7 + 5

√5

2

79.

√mn√

m+ 3√n

m√n− 3n

√n

m− 9n

80.√y√y − 3+

x2 + 8x√x− 45

√x

x(x− 9)

Solve the equation for the give variable.

81. 2√q + 4 = 5

q =9

4

82.√4r2 + r − 3 = 2r

r = 3

83. s−12 = 9

s =1

81

84. (t− 3)23 = −4

85. (u+ 2)4 = 32

u = 2 4√2± 2

86.√5r − r2 =

√6

r = 2, 3

87.√q − 3 =

√q + 2− 1

8

q = 7

88. 1 +√y + 7 =

√2y + 7

y = 9

89. 2√p−√p− 3 = 3

p = 4

90.√c−√c− 1 = 1

c = 1

91.√d+ 4− 2

√d− 1 = −1

d = 5

Simplify.

92. (2 + i)(3 + 4i)

2 + 11i

93. i25

i

94.6

7− 2i

42 + 12i

53

95. 2√−20−

√−45

i√5

96.−4 +

√−32

2

−2 + 2i√2

Complete the square.

97. q2 + 2q

9

q2 + 2q + 1

98. r2 + 14

r2 + 14r + 49

99. s2 − 3s

s2 − 3s+9

4

100. t2 − 5t

t2 − 5t+25

4

101. u2 +1

4u

u2 +1

4u+

1

64

102. v2 +3

2v

v2 +3

2v +

9

16

103. w2 +2

3w

w2 +2

3w +

1

9

104. p2 +6

5p

p2 +6

5p+

9

25

Factor each perfect square polynomial.

105. x2 + 8x+ 16

(x+ 4)2

106. x2 − 10x+ 25

(x− 5)2

107. x2 − 5x+25

4(x− 5

2

)2

10

108. x2 + x+1

4(x+

1

2

)2

109. y2 − 4

7y +

4

49(y − 2

7

)2

110. y2 − 6

5y +

9

25(y − 3

5

)2

111. y2 +3

5y +

9

100(y +

3

10

)2

112. y2 +3

2y +

9

16(y +

3

4

)2

Complete the square to solve:

113. q2 + 5q = 14

q = −7, 2

114. w2 − w − 20

w = −4, 5

115. 4x2 − 4x− 1 = 0

x =1±√2

2

116. y2 = −1

2

y = ±i√2

2

117. 5z2 − 4z + 1 = 0

z =2± i

5

11

118. c2 + 4c = 6

c = −2±√10

119. d2 + 6d− 8 = 0

d = −3±√17

120. a2 + 8a− 4 = 0

a = −4±√5

121. b2 + 10b− 3 = 0

b = −2±√10

Use the Quadratic Formula: (Rearrange the equation to use the quadratic formula ifneeded.)

122. −8q2 − 2q + 1 = 0

q = −1

2,1

4

123. −6z2 + 7z + 3 = 0

z = −1

3,3

2

124. 16x2 + 24x+ 9

x = −3

4

125.1

3t2 − t+

1

6= 0

t =3±√7

2

126.9(3x− 5)2

4= 1

x =13

9,17

9

127.1

4y2 + y = 1

y = −2± 2√2

12

128.1

3p2 +

1

2p =

1

3

p = −2, 12

129.25(2r + 1)2

9= 0

r = −1

2

130.m

3m− 4=

2

m+ 4

m = 1± i√7

131. n =7n− 4

12(n− 1)

n =1

4,4

3

Solve the problem that is quadratic in form(Think substitution)

132. (2x− 1)2 − 4(2x− 1) + 2 = 0

x =3±√2

2

133. x4 − 20x2 + 64 = 0

x = ±2,±4

134. x4 − 13x2 + 36 = 0

x = ±2,±3

135. x6 + 7x3 = −8

x = −2, 1

136. x6 − 28x3 + 27 = 0

x = 1, 3

137. x− 3x12 + 2 = 0

x = 1, 4

138. x23 + 4x

13 + 3 = 0

13

x = −27,−1

139. x12 − 5x

14 + 6 = 0

x = 16, 81

140. x−2 + x−1 − 6 = 0

x = −1

3,1

2

141. x−2 − 2x−1 = 8

x = −1

2,1

4

142.

(1

y − 1

)2

+

(1

y − 1

)= 6

y =2

3,3

2

143.

(1

w + 1

)2

− 2

(1

w + 1

)− 24 = 0

y = −5

6,−5

4

144. Which way does the graph open?

y = −3x2 + 4x+ 2

down

145. Which way does the graph open?

y = 2x2 + 4x+ 8

up145. Which way does the graph open?

y = (10− x)2

down146. Which way does the graph open?

y = 5x2 + 13x+ 24

up

14

147. Which way does the graph open?

y = (x+ 6)2

up148. Graph:

y = −2x2 + 5

149. Graph:

y = (x+ 3)2

150. Find the vertex.

y = x2 + 8x− 3

vertex = (−4,−19)

151. Find the vertex.

y = −3x2 + 18x− 7

vertex = (3, 20)

152. Find the vertex.

y = x2 − x+ 1

vertex =

(1

2,3

4

)153. Find the intercepts.

y = 16− x2

x− int = (−4, 0), (4, 0)

y − int = (0, 16)

154. Find the intercepts.

15

y = x2 − x− 6

x− int = (−2, 0), (3, 0)

y − int = (0,−6)

155. Find the intercepts.

y = −4x2 + 12x− 9

x− int = (3

2, 0)

y − int = (0,−9)

156. Determine the domain and the range.

f(x) = x+ 1

Domain: (−∞,∞)

Range: (−∞,∞)

157. Determine the domain and the range.

f(x) = 5− x

Domain: (−∞,∞)

Range: (−∞,∞)

158. Determine the domain and the range.

f(x) =√x− 2

Domain: [2,∞)

Range: [0,∞)

159.Determine the domain and the range.

f(x) =√x+ 4

Domain: [−4,∞)

Range: [0,∞)

16

160. Determine the domain and the range.

f(x) =√2x

Domain: [0,∞)

Range: [0,∞)

161. Determine the domain and the range.

f(x) =√2x− 4

Domain: [2,∞)

Range: [0,∞)

162. Let f(x) = 4x− 3 and g(x) = x2 − 2x .

Find the following.

a. (f + g)(x)b. (f − g)(x)c. (f · g)(x)d. (

f

g)(x)

e. (f + g)(3)f. (f − g)(−3)g. (f · g)(−1)h. (

f

g)(4)

a. x2 + 2x− 3b. −x2 + 6x− 3c. 4x3 − 11x2 + 6x

d.4x− 3

x2 − 2xe. 12f. −30g. −21h.

13

8

163. Let f(x) = 2x− 3 , g(x) = x2 + 3x , h(x) =x+ 3

2a. (fof)(x)

b. (gog)(x)c. (hoh)(x)d. (fog)(−2)

17

e. (gof)(1)f. (foh)(x)g. (hof)(x)h. (fofof)(x)

a. 4x− 9b. x4 + 6x3 + 12x2 + 9x

c.x+ 9

4d. −7e. −2f. xg. xh. 8x− 21

Find the inverse of the indicated function.

164. f(x) = 5x

f−1(x) =x

5

165. j(x) = x+ 7

j−1(x) = x− 7

166. r(x) = 2x− 8

r−1(x) =x+ 8

2

167. m(x) =2

x

m−1(x) =2

x

168. z(x) = 3√x− 4

z−1(x) = x3 + 4

169. n(x) =2

x+ 1

n−1(x) =2

x− 1

170. u(x) = 3√3x+ 7

u−1(x) =x3 − 7

3

18

171. h(x) =1− x

x+ 3

h−1(x) =1− 3x

x+ 1

172. g(x) =x+ 1

x− 2

g−1(x) =2x+ 1

x− 1

173. k(x) = 2 3√x− 1

k−1(x) =

(x+ 1

2

)3

174. If f−1(x) is truly an inverse of f(x) then fof−1(x) =1

x. True or False.

False . If f−1(x) is truly an inverse of f(x) then fof−1(x) = xSolve the equation.

175. 2x = 64

x = 6

176.

(1

3

)x

= 9

x = −2

177. 10x =1

100

x = −2

178. −32−x = −81

x = −2179. 42x−1 = 16

x =3

2

180.

(1

4

)3x

= 16

x = −2

3

Write each exponential equation as a logarithm equation and each logarithm equationas an exponential.

19

181. log2(8) = 3

23 = 8

182. logc(t) = 4

c4 = t

183. logb(N) = m

bm = N

184. log(100) = 2

102 = 100

185. e3 = x

ln(x) = 3

186. m = ex

ln(m) = x

187. 53 = 125

log5(125) = 3

188. log3(x) = 10

310 = x

189. a3 = c

loga(c) = 3

190. 2a = b

log2(b) = a

Simplify (Inverses)

191. log2(210)

10

20

192. ln(e9)

9

193. 5log5(15)

15

194. log(108)

8

195. eln(4)

4

196. 3log3(20)

20Write as as single logarithm and simplify. (Product Rule)

197. log(3) + log(7)

log(21)

198. ln(5) + ln(4)

ln(20)199. log3(

√5) + log3(

√x)

log3(√5x)

200. ln(a3) + ln(a5)

ln(a8)

201. ln(2) + ln(3) + ln(5)

ln(30)

202. log2(x) + log2(y) + log2(z)

log2(xyz)

203. log(x) + log(x+ 3)

21

log(x2 + 3x)

204. ln(x− 1) + ln(x+ 1)

ln(x2 − 1)

205. log3(x− 4) + log3(x− 5)

log3(x2 − 9x+ 20)

Write as as single logarithm and simplify. (Quotient Rule)

206. log(8)− log(2)

log(4)

207.ln(3)− ln(6)

ln

(1

2

)208. log2(x

6)− log2(x2)

log2(x4)

209. log3(x2 − 9)− log3(x− 3)

log3(x+ 3)

210. ln(x2 + x− 6)− ln(x+ 3)

ln(x− 2)

Write each expression in terms of log(3). (Power Rule)

211. log(27)

3log(3)

212. log

(1

9

)−2log(3)

213. log(√3)

22

1

2log(3)

214. log(3x)

xlog(3)

215. log(3100)

100log(3)

Solve the logarithmic equation.

216. log(x− 5) = 2

x = 105

217. log2(x+ 1) = 3

x = 7

218. 4log3(2x)− 1 = 7

x =9

2

219. ln(x) + ln(x+ 5) = ln(x+ 1) + ln(x+ 3)

3

220. log(x) + log(x+ 5) = 2 · log(x+ 2)

4

23