Question 1

Simplify the complex fraction.
-$$\left( a ^ { - 1 } + b ^ { - 1 } \right) ^ { - 1 }$$

Accepted Answer

A)  $\frac { a + b } { a b }$ 
B)  $\frac { a b } { a + b }$ 
C)  $\frac { a b } { a - b }$ 
D)  $\frac { a b } { a }$ 
A)  $\frac { a + b } { a b }$ 
B)  $\frac { a b } { a + b }$ 
C)  $\frac { a b } { a - b }$ 
D)  $\frac { a b } { a }$

Question 2

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { y ^ { 3 } - 64 } { y - 4 }$$

Accepted Answer

A)  $y ^ { 2 } - 16$ 
B)  $\frac { 1 } { y - 4 }$ 
C)  $y ^ { 2 } + 4 y + 16$ 
D)  Cannot be simplified 
A)  $y ^ { 2 } - 16$ 
B)  $\frac { 1 } { y - 4 }$ 
C)  $y ^ { 2 } + 4 y + 16$ 
D)  Cannot be simplified

Question 3

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 7 } { y ^ { 2 } - 1 }$$

Accepted Answer

A)  $\frac { 42 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
B)  $\frac { 10 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
C)  $\frac { 11 y - 10 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
D)  $\frac { 10 y - 11 } { ( y - 1 ) ( y - 2 ) }$ 
A)  $\frac { 42 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
B)  $\frac { 10 y - 11 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
C)  $\frac { 11 y - 10 } { ( y - 1 ) ( y + 1 ) ( y - 2 ) }$ 
D)  $\frac { 10 y - 11 } { ( y - 1 ) ( y - 2 ) }$

Question 4

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { 3 } { x } + \frac { 3 } { x - 4 }$$

Accepted Answer

A)  $\frac { 12 x - 6 } { x ( 4 - x ) }$ 
B)  $\frac { 6 x - 12 } { x ( x - 4 ) }$ 
C)  $\frac { 6 x - 12 } { x ( 4 - x ) }$ 
D)  $\frac { 12 x - 6 } { x ( x - 4 ) }$ 
A)  $\frac { 12 x - 6 } { x ( 4 - x ) }$ 
B)  $\frac { 6 x - 12 } { x ( x - 4 ) }$ 
C)  $\frac { 6 x - 12 } { x ( 4 - x ) }$ 
D)  $\frac { 12 x - 6 } { x ( x - 4 ) }$

Question 5

Subtract. Simplify the result, if possible.
-$\frac { 2 x ^ { 2 } } { x - 1 } - \frac { 2 x } { x - 1 }$

Accepted Answer

A)  $2 x$ 
B)  $\frac { 2 x } { x - 1 }$ 
C)  0 
D)  $\frac { 2 x ( x + 1 ) } { x - 1 }$ 
A)  $2 x$ 
B)  $\frac { 2 x } { x - 1 }$ 
C)  0 
D)  $\frac { 2 x ( x + 1 ) } { x - 1 }$

Question 6

Simplify the complex fraction.
-$$\frac { x ^ { - 2 } } { x ^ { - 2 } - y ^ { - 2 } }$$

Accepted Answer

A)  $\frac { y ^ { 2 } } { y ^ { 2 } - x ^ { 2 } }$ 
B)  $\frac { y } { y ^ { 2 } - x ^ { 2 } }$ 
C)  $\frac { y ^ { 2 } } { y ^ { 2 } + x ^ { 2 } }$ 
D)  $\frac { y ^ { 2 } - x ^ { 2 } } { y ^ { 2 } }$ 
A)  $\frac { y ^ { 2 } } { y ^ { 2 } - x ^ { 2 } }$ 
B)  $\frac { y } { y ^ { 2 } - x ^ { 2 } }$ 
C)  $\frac { y ^ { 2 } } { y ^ { 2 } + x ^ { 2 } }$ 
D)  $\frac { y ^ { 2 } - x ^ { 2 } } { y ^ { 2 } }$

Question 7

Perform the indicated operations. Simplify the result, if possible.
-$$\frac { x + 7 } { x ^ { 2 } - 3 x - 10 } + \frac { 3 x + 8 } { x ^ { 2 } - 11 x + 30 }$$

Accepted Answer

A)  $4 x + 15$ 
B)  $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x - 5 ) ( x + 2 ) ( x - 6 ) }$ 
C)  $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x + 5 ) ( x - 2 ) ( x + 6 ) }$ 
D)  $\frac { 4 x + 15 } { 2 x ^ { 2 } - 14 x + 20 }$ 
A)  $4 x + 15$ 
B)  $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x - 5 ) ( x + 2 ) ( x - 6 ) }$ 
C)  $\frac { 4 x ^ { 2 } + 15 x - 26 } { ( x + 5 ) ( x - 2 ) ( x + 6 ) }$ 
D)  $\frac { 4 x + 15 } { 2 x ^ { 2 } - 14 x + 20 }$

Question 8

Simplify the complex fraction.
-$$\frac { m ^ { - 1 } + z ^ { - 1 } } { m ^ { - 1 } - z ^ { - 1 } }$$

Accepted Answer

A)  $\frac { z + m } { z - m }$ 
B)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { m } }$ 
C)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { z } }$ 
D)  $\frac { \mathrm { z } - \mathrm { m } } { \mathrm { z } }$ 
A)  $\frac { z + m } { z - m }$ 
B)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { m } }$ 
C)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { z } }$ 
D)  $\frac { \mathrm { z } - \mathrm { m } } { \mathrm { z } }$

Question 9

Divide as indicated.
-$$\frac { 3 x - 3 } { x } \div \frac { 4 x - 4 } { 2 x ^ { 2 } }$$

Accepted Answer

A)  $\frac { 6 x ^ { 3 } - 6 x ^ { 2 } } { 4 x ^ { 2 } - 4 x }$ 
B)  $\frac { 3 x } { 2 }$ 
C)  $\frac { 2 } { 3 x }$ 
D)  $\frac { 12 x ^ { 2 } + 24 x + 12 } { 2 x ^ { 3 } }$ 
A)  $\frac { 6 x ^ { 3 } - 6 x ^ { 2 } } { 4 x ^ { 2 } - 4 x }$ 
B)  $\frac { 3 x } { 2 }$ 
C)  $\frac { 2 } { 3 x }$ 
D)  $\frac { 12 x ^ { 2 } + 24 x + 12 } { 2 x ^ { 3 } }$

Question 10

Multiply as indicated.
-$$\frac { x ^ { 2 } + 6 x + 8 } { x ^ { 2 } + 8 x + 12 } \cdot \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 11 x + 28 }$$

Accepted Answer

A)  $\frac { x } { x + 7 }$ 
B)  $\frac { x ^ { 2 } + 6 x } { x + 7 }$ 
C)  $\frac { x } { x ^ { 2 } + 8 x + 12 }$ 
D)  $\frac { 1 } { x + 7 }$ 
A)  $\frac { x } { x + 7 }$ 
B)  $\frac { x ^ { 2 } + 6 x } { x + 7 }$ 
C)  $\frac { x } { x ^ { 2 } + 8 x + 12 }$ 
D)  $\frac { 1 } { x + 7 }$

Question 11

Find the domain of the rational function.
-$$f ( x ) = \frac { 4 } { x + 4 }$$

Accepted Answer

A)  domain of $f : ( \infty , - 4 ) \cup ( - 4 , \infty )$ 
B)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$ 
C)  domain of $\mathrm { f } : ( \infty , 4 ) \cup ( 4 , \infty )$ 
D)  domain of $\mathrm { f } : ( \infty , \infty )$ 
A)  domain of $f : ( \infty , - 4 ) \cup ( - 4 , \infty )$ 
B)  domain of $f : ( \infty , 0 ) \cup ( 0 , \infty )$ 
C)  domain of $\mathrm { f } : ( \infty , 4 ) \cup ( 4 , \infty )$ 
D)  domain of $\mathrm { f } : ( \infty , \infty )$

Question 12

Solve the problem.
-A company that produces radios has costs given by the function C(x) = 25x + 25,000, where x is the number of radios manufactured and C(x) is measured in dollars. The average cost to manufacture each radio is given by $\bar { C } ( x ) = \frac { 25 x + 25,000 } { x }$. Find $\bar { C } ( 50 )$. (Round to the nearest dollar, if necessary.)

Accepted Answer

A)  $\$ 75$ 
B)  $\$ 525$ 
C)  $\$ 74$ 
D)  $\$ 535$ 
A)  $\$ 75$ 
B)  $\$ 525$ 
C)  $\$ 74$ 
D)  $\$ 535$

Question 13

Solve the problem.
-Suppose a cost-benefit model is given by $y = \frac { 5.3 x } { 100 - x }$, where $y$ is the cost in thousands of dollars for removing $x$ percent of a given pollutant. Find the cost for removing $60 \%$ of the pollutant.

Accepted Answer

A)  $\$ 79.5$ thousands 
B)  $\$ 9.2$ thousands 
C)  $\$ 8$ thousands 
D)  $\$ 2$ thousands 
A)  $\$ 79.5$ thousands 
B)  $\$ 9.2$ thousands 
C)  $\$ 8$ thousands 
D)  $\$ 2$ thousands

Question 14

Perform the indicated operations. Simplify the result, if possible.
-$\frac { 2 } { x ^ { 2 } } - \frac { 8 } { x }$

Accepted Answer

A)  $\frac { 8 x - 2 } { x }$ 
B)  $\frac { 2 - 8 x } { x ^ { 2 } }$ 
C)  $\frac { 2 + 8 x } { x ^ { 2 } }$ 
D)  $\frac { 2 x + 8 } { x ^ { 2 } }$ 
A)  $\frac { 8 x - 2 } { x }$ 
B)  $\frac { 2 - 8 x } { x ^ { 2 } }$ 
C)  $\frac { 2 + 8 x } { x ^ { 2 } }$ 
D)  $\frac { 2 x + 8 } { x ^ { 2 } }$

Question 15

Solve the problem.
-To calculate the drug dosage for a child, a pharmacist may use the formula $\mathrm { d } ( \mathrm { x } ) = \frac { \mathrm { Dx } } { \mathrm { x } + 10 } , 0 \leq \mathrm { x } \leq 12$. The child's age is $x$ and the adult dosage is D. What is the dosage for an 3 -year old child if the adult dosage is 40 mg? (Round to the nearest tenth.)

Accepted Answer

A)  $3.1 \mathrm { mg }$ 
B)  $12.0 \mathrm { mg }$ 
C)  $9.2 \mathrm { mg }$ 
D)  $40.0 \mathrm { mg }$ 
A)  $3.1 \mathrm { mg }$ 
B)  $12.0 \mathrm { mg }$ 
C)  $9.2 \mathrm { mg }$ 
D)  $40.0 \mathrm { mg }$

Question 16

Multiply as indicated.
-$$\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 5 x } { - 50 x - 50 }$$

Accepted Answer

A)  $- \frac { x ^ { 2 } + 1 } { 10 }$ 
B) $ = \frac { x ^ { 3 } + 1 } { 10 ( x + 1 ) }$ 
C)  $- \frac { 1 } { 10 }$ 
D)  $\frac { x + 1 } { 10 ( - x - 1 ) }$ 
A)  $- \frac { x ^ { 2 } + 1 } { 10 }$ 
B) $ = \frac { x ^ { 3 } + 1 } { 10 ( x + 1 ) }$ 
C)  $- \frac { 1 } { 10 }$ 
D)  $\frac { x + 1 } { 10 ( - x - 1 ) }$

Question 17

Add. Simplify the result, if possible.
-$$\frac { x ^ { 2 } - 5 x } { x - 2 } + \frac { 6 } { x - 2 }$$

Accepted Answer

A)  $x + 3$ 
B)  $\frac { x ^ { 2 } - 5 x + 6 } { 2 }$ 
C)  $x - 3$ 
D)  $x - 2$ 
A)  $x + 3$ 
B)  $\frac { x ^ { 2 } - 5 x + 6 } { 2 }$ 
C)  $x - 3$ 
D)  $x - 2$

Question 18

Add. Simplify the result, if possible.
-$\frac { 4 } { 9 x ^ { 2 } } + \frac { 2 } { 9 x ^ { 2 } }$

Accepted Answer

A)  $\frac { 2 } { 3 x ^ { 4 } }$ 
B)  2 
C)  $\frac { 2 } { 3 x ^ { 2 } }$ 
D)  $\frac { 3 } { 2 x ^ { 2 } }$ 
A)  $\frac { 2 } { 3 x ^ { 4 } }$ 
B)  2 
C)  $\frac { 2 } { 3 x ^ { 2 } }$ 
D)  $\frac { 3 } { 2 x ^ { 2 } }$

Question 19

Add. Simplify the result, if possible.
-$$\frac { x } { x + 5 } + \frac { - 8 } { x + 5 }$$

Accepted Answer

A)  $- \frac { 8 } { 5 }$ 
B)  $\frac { - 8 x } { x ^ { 2 } + 10 x + 25 }$ 
C)  $\frac { x - 8 } { x + 5 }$ 
D)  $\frac { - 7 x } { x + 5 }$ 
A)  $- \frac { 8 } { 5 }$ 
B)  $\frac { - 8 x } { x ^ { 2 } + 10 x + 25 }$ 
C)  $\frac { x - 8 } { x + 5 }$ 
D)  $\frac { - 7 x } { x + 5 }$

Question 20

Simplify the complex fraction.
-$$\frac { - 8 x ^ { - 1 } + ( 7 y ) ^ { - 1 } } { x ^ { - 2 } }$$

Accepted Answer

A)  $\frac { x ^ { 2 } } { - 8 x + 7 y }$ 
B)  $\frac { 7 y } { - 56 x y + x ^ { 2 } }$ 
C)  $\frac { - 56 x y + x ^ { 2 } } { 7 y }$ 
D)  $\frac { - 8 x y + 7 x ^ { 2 } } { y }$ 
A)  $\frac { x ^ { 2 } } { - 8 x + 7 y }$ 
B)  $\frac { 7 y } { - 56 x y + x ^ { 2 } }$ 
C)  $\frac { - 56 x y + x ^ { 2 } } { 7 y }$ 
D)  $\frac { - 8 x y + 7 x ^ { 2 } } { y }$

Simplify the complex fraction. - $\left( a ^ { - 1 } + b ^ { - 1 } \right) ^ { - 1 }$

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { y ^ { 3 } - 64 } { y - 4 }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 7 } { y ^ { 2 } - 1 }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { 3 } { x } + \frac { 3 } { x - 4 }$

Subtract. Simplify the result, if possible. - $\frac { 2 x ^ { 2 } } { x - 1 } - \frac { 2 x } { x - 1 }$

Simplify the complex fraction. - $\frac { x ^ { - 2 } } { x ^ { - 2 } - y ^ { - 2 } }$

Perform the indicated operations. Simplify the result, if possible. - $\frac { x + 7 } { x ^ { 2 } - 3 x - 10 } + \frac { 3 x + 8 } { x ^ { 2 } - 11 x + 30 }$

Simplify the complex fraction. - $\frac { m ^ { - 1 } + z ^ { - 1 } } { m ^ { - 1 } - z ^ { - 1 } }$

Divide as indicated. - $\frac { 3 x - 3 } { x } \div \frac { 4 x - 4 } { 2 x ^ { 2 } }$

Multiply as indicated. - $\frac { x ^ { 2 } + 6 x + 8 } { x ^ { 2 } + 8 x + 12 } \cdot \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 11 x + 28 }$

$Find the domain of the rational function. - f ( x ) = \frac { 4 } { x + 4 }$ Find the domain of the rational function. - $f ( x ) = \frac { 4 } { x + 4 }$

Solve the problem. -Suppose a cost-benefit model is given by $y = \frac { 5.3 x } { 100 - x }$ , where $y$ is the cost in thousands of dollars for removing $x$ percent of a given pollutant. Find the cost for removing $60 \%$ of the pollutant.

Perform the indicated operations. Simplify the result, if possible. - $\frac { 2 } { x ^ { 2 } } - \frac { 8 } { x }$

Multiply as indicated. - $\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 5 x } { - 50 x - 50 }$

Add. Simplify the result, if possible. - $\frac { x ^ { 2 } - 5 x } { x - 2 } + \frac { 6 } { x - 2 }$

Add. Simplify the result, if possible. - $\frac { 4 } { 9 x ^ { 2 } } + \frac { 2 } { 9 x ^ { 2 } }$

Add. Simplify the result, if possible. - $\frac { x } { x + 5 } + \frac { - 8 } { x + 5 }$

Simplify the complex fraction. - $\frac { - 8 x ^ { - 1 } + ( 7 y ) ^ { - 1 } } { x ^ { - 2 } }$

Algebra, Mathematical Models, and Problem Solving

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Exam 6: Rational Expressions, Functions, and Equations

Simplify the complex fraction. - (a−1+b−1)−1\left( a ^ { - 1 } + b ^ { - 1 } \right) ^ { - 1 }(a−1+b−1)−1

Simplify the rational expression. If the rational expression cannot be simplified, so state. - y3−64y−4\frac { y ^ { 3 } - 64 } { y - 4 }y−4y3−64​

Perform the indicated operations. Simplify the result, if possible. - 3y2−3y+2+7y2−1\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 7 } { y ^ { 2 } - 1 }y2−3y+23​+y2−17​

Perform the indicated operations. Simplify the result, if possible. - 3x+3x−4\frac { 3 } { x } + \frac { 3 } { x - 4 }x3​+x−43​

Subtract. Simplify the result, if possible. - 2x2x−1−2xx−1\frac { 2 x ^ { 2 } } { x - 1 } - \frac { 2 x } { x - 1 }x−12x2​−x−12x​

Simplify the complex fraction. - x−2x−2−y−2\frac { x ^ { - 2 } } { x ^ { - 2 } - y ^ { - 2 } }x−2−y−2x−2​

Perform the indicated operations. Simplify the result, if possible. - x+7x2−3x−10+3x+8x2−11x+30\frac { x + 7 } { x ^ { 2 } - 3 x - 10 } + \frac { 3 x + 8 } { x ^ { 2 } - 11 x + 30 }x2−3x−10x+7​+x2−11x+303x+8​

Simplify the complex fraction. - m−1+z−1m−1−z−1\frac { m ^ { - 1 } + z ^ { - 1 } } { m ^ { - 1 } - z ^ { - 1 } }m−1−z−1m−1+z−1​

Divide as indicated. - 3x−3x÷4x−42x2\frac { 3 x - 3 } { x } \div \frac { 4 x - 4 } { 2 x ^ { 2 } }x3x−3​÷2x24x−4​

Multiply as indicated. - x2+6x+8x2+8x+12⋅x2+6xx2+11x+28\frac { x ^ { 2 } + 6 x + 8 } { x ^ { 2 } + 8 x + 12 } \cdot \frac { x ^ { 2 } + 6 x } { x ^ { 2 } + 11 x + 28 }x2+8x+12x2+6x+8​⋅x2+11x+28x2+6x​

Find the domain of the rational function. - f(x)=4x+4f ( x ) = \frac { 4 } { x + 4 }f(x)=x+44​

Solve the problem. -Suppose a cost-benefit model is given by y=5.3x100−xy = \frac { 5.3 x } { 100 - x }y=100−x5.3x​ , where yyy is the cost in thousands of dollars for removing xxx percent of a given pollutant. Find the cost for removing 60%60 \%60% of the pollutant.

Perform the indicated operations. Simplify the result, if possible. - 2x2−8x\frac { 2 } { x ^ { 2 } } - \frac { 8 } { x }x22​−x8​

Multiply as indicated. - x3+1x3−x2+x⋅5x−50x−50\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 5 x } { - 50 x - 50 }x3−x2+xx3+1​⋅−50x−505x​

Add. Simplify the result, if possible. - x2−5xx−2+6x−2\frac { x ^ { 2 } - 5 x } { x - 2 } + \frac { 6 } { x - 2 }x−2x2−5x​+x−26​

Add. Simplify the result, if possible. - 49x2+29x2\frac { 4 } { 9 x ^ { 2 } } + \frac { 2 } { 9 x ^ { 2 } }9x24​+9x22​

Add. Simplify the result, if possible. - xx+5+−8x+5\frac { x } { x + 5 } + \frac { - 8 } { x + 5 }x+5x​+x+5−8​

Simplify the complex fraction. - −8x−1+(7y)−1x−2\frac { - 8 x ^ { - 1 } + ( 7 y ) ^ { - 1 } } { x ^ { - 2 } }x−2−8x−1+(7y)−1​