Download MTH_112_Exam_1_201340_Solutions

MTH 112
Exam 1
Prof. Townsend
Fall 2013
Problems
1) Simplify the expression. Use positive exponents. Assume variables represent nonnegative
numbers.
−18a 4 b 6 2a
= 2
✓
−9a 3b 8
b
−15a 7b 3 5a 4
= 5
−3a 3b 8
b
✓
−10a 5b 6
= 2a 2b ✓
3 5
−5a b
−14a 4 b 2 2a 3
= 6
−7ab 8
b
✓
2) Evaluate the given expression by performing the indicated operations.
−12
12
4
2
−8 −
− 4 ( 6 − 9 ) = −8 +
− 4 ( −3) = 4 − = 2 ✓
3( 2 − 5 )
−9
3
3
−7 −
−12
4
4
1
− 4 ( 3 − 9 ) = −7 + 24 − = 17 − = 16
3( 2 − 7 )
5
5
5
−6 −
−12
− 4 ( 5 − 9 ) = 16 − 6 − 1 = 9 ✓
3( 2 − 6 )
−3 −
−12
− 4 ( 6 − 7 ) = −3 + 4 − 1 = 0 ✓
3(1− 5 )
Page 1 of 6
✓
3) Simplify the given expression. Express results with positive exponents only.
⎛ −3b 2 ⎞
⎜⎝ z 5 ⎟⎠
−3
⎛ −3a 2 ⎞
⎜⎝ u 2 ⎟⎠
−3
⎛ −4c 2 ⎞
⎜⎝ t 5 ⎟⎠
−5
⎛ −5 f 2 ⎞
⎜⎝ g 5 ⎟⎠
−3
3
⎛ z5 ⎞
z15
=⎜
=− 3 6
3b
⎝ −3b 2 ⎟⎠
✓
3
⎛ u2 ⎞
u6
=⎜
=− 3 6
3a
⎝ −3a 2 ⎟⎠
✓
5
⎛ t5 ⎞
t 25
=⎜
=
−
4 5 c10
⎝ −4c 2 ⎟⎠
✓
3
⎛ g5 ⎞
g15
=⎜
=
−
53 f 6
⎝ −5 f 2 ⎟⎠
✓
4) Factor the following expression completely.
5a 4 − 125a 2 = 5a 2 ( a 2 − 25 ) = 5a 2 ( a − 5 ) ( a + 5 ) ✓
81− y 4 = 34 − y 4 = ( 32 + y 2 ) ( 32 − y 2 ) = ( 32 + y 2 ) ( 3 + y ) ( 3 − y ) ✓
x 4 − 5x 2 − 36 = ( x 2 + 4 ) ( x 2 − 9 ) = ( x 2 + 4 ) ( x + 3) ( x − 3) ✓
3x 4 − 33x 2 + 30 = 3( x 4 − 11x 2 + 10 ) = 3( x 2 − 10 ) ( x 2 − 1) = 3( x 2 − 10 ) ( x − 1) ( x + 1) ✓
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5) Simplify the following expression. Be sure to factor completely.
3ax ( x − 3) x ( 2x + 1) 3ax 2 3x
3ax 2 − 9ax
2x 2 + x
✓
×
=
×
=
=
10x 2 + 5x a 2 x − 3a 2 5x ( 2x + 1) a 2 ( x − 3) 5a 2 x 5a
a 2 − a a 2 + 2a + 1 a 2 − a 18 − 2a 2
÷
=
×
=
3a + 9 18 − 2a 2
3a + 9 a 2 + 2a + 1
2
2
a ( a − 1) 2 ( 3 − a ) a ( a − 1) 2 ( 3 + a ) ( 3 − a ) 2a ( a − 1) ( 3 − a ) ✓
=
×
=
×
=
2
3( a + 3)
3( a + 1)
( a + 1)2 3( a + 3)
( a + 1)2
2
2
4x 2 − 36 7x − 35 4 ( x − 3 ) 7 ( x − 5 )
×
=
×
=
x 3 − 25x 3x 2 + 9x x ( x 2 − 5 2 ) 3x ( x + 3)
=
4 ( x − 3) ( x + 3) 7 ( x − 5 )
28 ( x − 3)
✓
×
= 2
x ( x − 5 ) ( x + 5 ) 3x ( x + 3) 3x ( x + 5 )
x 2 + 2x + 1 x 2 − x x 2 + 2x + 1 3x + 9
( x + 1) × 3( x + 3) =
÷
=
× 2
=
2
2
18 − 2x
3x + 9
18 − 2x
x − x 2 ( 32 − x 2 ) x ( x − 1)
2
2
2
x + 1)
3( x + 3)
3( x + 1)
(
=
×
=
2 ( 3 + x ) ( 3 − x ) x ( x − 1) 2x ( 3 − x ) ( x − 1)
✓
6) Simplify the given algebraic expression.
9x − ⎡⎣ 6 − ( x − 2 ) + 5x ⎤⎦ = 9x − [ 6 + 5x − x + 2 ] = 9x − [ 8 + 4x ] = 5x − 8 ✓
7x − ⎡⎣ 3 − ( x − 4 ) + 5x ⎤⎦ = 7x − [ 3 + 5x − x + 4 ] = 7x − [ 7 + 4x ] = 3x − 7 ✓
5x − ⎡⎣ 6 − ( 2x − 5 ) + 7x ⎤⎦ = 5x − [ 6 + 7x − 2x + 5 ] = 5x − [11+ 5x ] = −11 ✓
9x − ⎡⎣ 5 − ( x − 3) + 6x ⎤⎦ = 9x − [ 5 + 6x − x + 3] = 9x − [ 8 + 5x ] = 4x − 8
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✓
7) Perform the indicated multiplications completely. There should be no parentheses in your
answer.
−3 3xw 4 2x − 3w 2 = −9xw 4 2x − 3w 2 = −18x 2 w 4 + 27xw 6 ✓
(
)(
)
(
)
−2 ( 4xw 3 ) ( 7x − 3w 4 ) = −8xw 3 ( 7x − 3w 4 ) = −56x 2 w 3 + 24xw 7 ✓
−5 ( 3xw 4 ) ( 2x 2 − 3w ) = −15xw 4 ( 2x 2 − 3w ) = −30x 3w 4 + 45xw 5 ✓
−2 ( 4xw 3 ) ( 2x − 2w 5 ) = −8xw 3 ( 2x − 2w 5 ) = −16x 2 w 3 + 16xw 8
✓
8) Factor the following expression completely. Be sure to show at least one intermediate step.
2ab 3 − 6a 2b + 8a 3b 3 = 2ab ( b 2 − 3a + 4a 2b 2 )
✓
16ab 3 − 4a 2b + 8a 3b 3 = 4ab ( 4b 2 − a + 2a 2b 2 )
✓
9ab 3 + 15a 2b − 3a 3b 3 = 3ab ( 3b 2 + 5a − a 2b 2 )
✓
6a 2b + 10ab 3 − 8a 3b 3 = 2ab ( 3b 2 + 5a − 4a 2b 2 ) ✓
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9) Solve the given equation for x.
4 − 2 ( 6 − 7x )
6x = 4 − 2 ( 6 − 7x ) = 4 − 12 + 14x = 14x − 8
2x =
3
−8x = −8 x = 1 ✓
3x =
5 − 2 ( 6 − 7x )
2
6x = 5 − 2 ( 6 − 7x ) = 5 − 12 + 14x = −7 + 14x
−8x = −7 x =
7
✓
8
2x =
7 − 2 ( 6 − 2x )
3
6x = 7 − 2 ( 6 − 2x ) = 7 − 12 + 4x = 4x − 5
2x = −5 x = −
5
✓
2
2x =
3 − 2 ( 4 − 7x )
3
6x = 3 − 2 ( 4 − 7x ) = 3 − 8 + 14x = 14x − 5
10) Solve the given equation for x.
u = u0 − abx
u − u0 = −abx
z = z0 − abx
z − z0 = −abx
y = y0 − cdx
y − y0 = −cdx
w = w0 − adx
w − w0 = −adx
u − u0
ab
z − z0
x=−
ab
y − y0
x=−
cd
w − w0
x=−
ad
x=−
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✓
✓
✓
✓
−8x = −5 x =
5
✓
8
Page 6 of 6