Question 1

Divide. Simplify if possible.
-$$\frac { 3 y - 3 } { y } \div \frac { 5 y - 5 } { 8 y ^ { 2 } }$$

Accepted Answer

A)  $\frac { 24 y ^ { 3 } - 24 y ^ { 2 } } { 5 y ^ { 2 } - 5 y }$ 
B)  $\frac { 5 } { 24 y }$ 
C)  $\frac { 24 y } { 5 }$ 
D)  $\frac { 15 y ^ { 2 } + 30 y + 15 } { 8 y ^ { 3 } }$ 
A)  $\frac { 24 y ^ { 3 } - 24 y ^ { 2 } } { 5 y ^ { 2 } - 5 y }$ 
B)  $\frac { 5 } { 24 y }$ 
C)  $\frac { 24 y } { 5 }$ 
D)  $\frac { 15 y ^ { 2 } + 30 y + 15 } { 8 y ^ { 3 } }$

Question 2

Provide an appropriate response.
-A formula for the focal length of a lens is $$f = \frac { a b } { b + a }$$ Calculate f (the focal length) for a = 8 cm and b = 17 cm. Round to the nearest tenth.

Accepted Answer

A)  2.6 cm 
B)  4.0 cm 
C)  8.5 cm 
D)  5.4 cm 
A)  2.6 cm 
B)  4.0 cm 
C)  8.5 cm 
D)  5.4 cm

Question 3

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { 3 x ^ { 2 } + 9 x ^ { 3 } } { 2 x + 6 x ^ { 2 } }$$

Accepted Answer

A)  $\frac { 3 x } { 2 }$ 
B)  cannot be simplified 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 3 + 9 x ^ { 3 } } { 2 x + 6 }$ 
A)  $\frac { 3 x } { 2 }$ 
B)  cannot be simplified 
C)  $\frac { 3 } { 2 }$ 
D)  $\frac { 3 + 9 x ^ { 3 } } { 2 x + 6 }$

Question 4

Perform the indicated operation. Simplify if possible.
-$$\frac { 3 } { x + 13 } + \frac { 4 } { x + 13 }$$

Accepted Answer

A)  $\frac { 7 } { x + 13 }$ 
B)  $\frac { x + 13 } { 7 }$ 
C)  $\frac { 7 } { x + 26 }$ 
D)  1 
A)  $\frac { 7 } { x + 13 }$ 
B)  $\frac { x + 13 } { 7 }$ 
C)  $\frac { 7 } { x + 26 }$ 
D)  1

Question 5

Perform the indicated operation. Simplify if possible.
-A square shaped pasture has a side of length x 9+ 7 meters.  
Express the perimeter of the pasture as a rational expression.

Accepted Answer

A)  $\frac { 36 } { x + 14 } m$ 
B)  $\frac { 9 } { x + 28 } m$ 
C)  $\frac { 36 } { x + 28 } m$ 
D)  $\frac { 36 } { x + 7 } m$ 
A)  $\frac { 36 } { x + 14 } m$ 
B)  $\frac { 9 } { x + 28 } m$ 
C)  $\frac { 36 } { x + 28 } m$ 
D)  $\frac { 36 } { x + 7 } m$

Question 6

Divide. Simplify if possible.
-$$\frac { x ^ { 2 } + 16 x + 63 } { x ^ { 2 } + 18 x + 81 } + \frac { x ^ { 2 } + 7 x } { x ^ { 2 } + 2 x - 63 }$$

Accepted Answer

A)  $\frac { x } { x ^ { 2 } + 18 x + 81 }$ 
B)  $\frac { x - 7 } { x ^ { 2 } + 9 x }$ 
C)  $x - 7$ 
D)  $\frac { x - 7 } { x }$ 
A)  $\frac { x } { x ^ { 2 } + 18 x + 81 }$ 
B)  $\frac { x - 7 } { x ^ { 2 } + 9 x }$ 
C)  $x - 7$ 
D)  $\frac { x - 7 } { x }$

Question 7

Perform the indicated operation. Simplify if possible.
-$$\frac { 3 y ^ { 2 } } { y - 1 } + \frac { - 3 y } { y - 1 }$$

Accepted Answer

A)  0 
B)  $\frac { 3 y ( y + 1 ) } { y - 1 }$ 
C)  $3 y$ 
D)  $\frac { 3 y } { y - 1 }$ 
A)  0 
B)  $\frac { 3 y ( y + 1 ) } { y - 1 }$ 
C)  $3 y$ 
D)  $\frac { 3 y } { y - 1 }$

Question 8

Simplify the rational expression. If the rational expression cannot be simplified, so state.
-$$\frac { y ^ { 2 } - 10 y + 24 } { 4 - y }$$

Accepted Answer

A)  $- y + 6$ 
B)  $y + 6$ 
C)  $- y - 6$ 
D)  $y - 6$ 
A)  $- y + 6$ 
B)  $y + 6$ 
C)  $- y - 6$ 
D)  $y - 6$

Question 9

Multiply. Simplify if possible.
-$$\frac { 2 p - 2 } { p } \cdot \frac { 7 p ^ { 2 } } { 6 p - 6 }$$

Accepted Answer

A)  $\frac { 12 p ^ { 2 } + 24 p + 12 } { 7 p ^ { 3 } }$ 
B)  $\frac { 14 p ^ { 3 } - 14 p ^ { 2 } } { 6 p ^ { 2 } - 6 p }$ 
C)  $\frac { 7 p } { 3 }$ 
D)  $\frac { 3 } { 7 \mathrm { p } }$ 
A)  $\frac { 12 p ^ { 2 } + 24 p + 12 } { 7 p ^ { 3 } }$ 
B)  $\frac { 14 p ^ { 3 } - 14 p ^ { 2 } } { 6 p ^ { 2 } - 6 p }$ 
C)  $\frac { 7 p } { 3 }$ 
D)  $\frac { 3 } { 7 \mathrm { p } }$

Question 10

Add or subtract as indicated. Simplify the result, if possible.
-$$\frac { 5 - x } { x - 6 } - \frac { 2 x + 6 } { 6 - x }$$

Accepted Answer

A)  $- \frac { x - 1 } { x - 6 }$ 
B)  $- \frac { x + 11 } { x - 6 }$ 
C)  $\frac { x + 11 } { x - 6 }$ 
D)  $\frac { x - 1 } { x - 6 }$ 
A)  $- \frac { x - 1 } { x - 6 }$ 
B)  $- \frac { x + 11 } { x - 6 }$ 
C)  $\frac { x + 11 } { x - 6 }$ 
D)  $\frac { x - 1 } { x - 6 }$

Question 11

Perform the indicated operation. Simplify if possible.
-$$\frac { 5 x + 4 } { 10 } + \frac { 7 x - 4 } { 10 }$$

Accepted Answer

A)  $- \frac { 6 x } { 5 }$ 
B)  $\frac { 6 x } { 5 }$ 
C)  $\frac { 12 x } { 5 }$ 
D)  $\frac { 35 x } { 5 }$ 
A)  $- \frac { 6 x } { 5 }$ 
B)  $\frac { 6 x } { 5 }$ 
C)  $\frac { 12 x } { 5 }$ 
D)  $\frac { 35 x } { 5 }$

Question 12

Perform the indicated operation(s). Simplify if possible.
-$\frac { 6 } { 9 x ^ { 2 } } - \frac { 2 } { 9 x ^ { 2 } }$

Accepted Answer

A)  $\frac { 4 } { 9 x ^ { 2 } }$ 
B)  $\frac { 9 } { 4 x ^ { 2 } }$ 
C)  $\frac { 4 } { 18 x ^ { 4 } }$ 
D)  4 
A)  $\frac { 4 } { 9 x ^ { 2 } }$ 
B)  $\frac { 9 } { 4 x ^ { 2 } }$ 
C)  $\frac { 4 } { 18 x ^ { 4 } }$ 
D)  4

Question 13

Find the missing expression.
-$$\frac { 13 x - 3 } { x - 5 } - \frac { ? } { 5 - x } = \frac { 22 x + 1 } { x - 5 }$$

Accepted Answer

A)  $35 x + 4$ 
B)  $9 x + 4$ 
C)  $- 9 x - 4$ 
D)  $9 x - 4$ 
A)  $35 x + 4$ 
B)  $9 x + 4$ 
C)  $- 9 x - 4$ 
D)  $9 x - 4$

Question 14

Perform the indicated operation. Simplify if possible.
-$$\frac { x ^ { 2 } - 8 x } { x - 6 } + \frac { 12 } { x - 6 }$$

Accepted Answer

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

Question 15

Divide. Simplify if possible.
-$$\frac { x ^ { 2 } - 2 x + x y - 2 y } { 9 x ^ { 2 } - 9 y ^ { 2 } } \div \frac { x - 2 } { 12 x - 12 y }$$

Accepted Answer

A)  $\frac { 12 \left( x ^ { 2 } - 2 x + x y - 2 y \right) } { 9 ( x + y ) ( x - 2 ) }$ 
B)  1 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 108 ( x - y ) ^ { 2 } }$ 
A)  $\frac { 12 \left( x ^ { 2 } - 2 x + x y - 2 y \right) } { 9 ( x + y ) ( x - 2 ) }$ 
B)  1 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 108 ( x - y ) ^ { 2 } }$

Question 16

Multiply. Simplify if possible.
-$$\frac { k ^ { 2 } + 13 k + 36 } { k ^ { 2 } + 11 k + 18 } \cdot \frac { k ^ { 2 } + 4 k + 4 } { k ^ { 2 } + 6 k + 8 }$$

Accepted Answer

A)  1 
B)  $\frac { k + 4 } { k + 2 }$ 
C)  $\frac { k + 2 } { k + 2 }$ 
D)  $\frac { 1 } { k + 2 }$ 
A)  1 
B)  $\frac { k + 4 } { k + 2 }$ 
C)  $\frac { k + 2 } { k + 2 }$ 
D)  $\frac { 1 } { k + 2 }$

Question 17

Multiply. Simplify if possible.
-$$\frac { ( y - w ) ^ { 2 } } { y + w } \cdot \frac { y } { y ^ { 2 } - y w }$$

Accepted Answer

A)  $\frac { y - w } { y + w }$ 
B)  1 
C)  $\frac { w } { y + w }$ 
D)  $\frac { y + w } { y - w }$ 
A)  $\frac { y - w } { y + w }$ 
B)  1 
C)  $\frac { w } { y + w }$ 
D)  $\frac { y + w } { y - w }$

Question 18

Perform the indicated operation(s). Simplify if possible.
-$$\frac { 2 x + 7 } { x + 8 } - \frac { 3 x + 5 } { x + 8 } - \frac { 9 x } { x + 8 }$$

Accepted Answer

A)  $\frac { x + 2 } { x + 24 }$ 
B)  $\frac { - 14 x + 12 } { x + 8 }$ 
C)  $\frac { - 10 x + 2 } { 3 x + 24 }$ 
D)  $\frac { - 10 x + 2 } { x + 8 }$ 
A)  $\frac { x + 2 } { x + 24 }$ 
B)  $\frac { - 14 x + 12 } { x + 8 }$ 
C)  $\frac { - 10 x + 2 } { 3 x + 24 }$ 
D)  $\frac { - 10 x + 2 } { x + 8 }$

Question 19

Perform the indicated operation(s). Simplify if possible.
-$$\frac { 24 } { x - 8 } - \frac { 6 } { x - 8 }$$

Accepted Answer

A)  $\frac { 24 ( x - 8 ) } { 6 ( x - 8 ) }$ 
B)  $\frac { 18 } { x }$ 
C)  $\frac { 18 } { x - 8 }$ 
D)  $\frac { 30 } { x - 8 }$ 
A)  $\frac { 24 ( x - 8 ) } { 6 ( x - 8 ) }$ 
B)  $\frac { 18 } { x }$ 
C)  $\frac { 18 } { x - 8 }$ 
D)  $\frac { 30 } { x - 8 }$

Question 20

Find the least common denominator of the rational expressions.
-$\frac { 6 } { 6 y + 42 }$ and $\frac { 3 } { y ^ { 2 } + 7 y }$

Accepted Answer

A)  $6 y + 7$ 
B)  $6 y ( y + 7 )$ 
C)  $6 y ^ { 2 } + 7$ 
D)  $6 y ^ { 2 } + 42$ 
A)  $6 y + 7$ 
B)  $6 y ( y + 7 )$ 
C)  $6 y ^ { 2 } + 7$ 
D)  $6 y ^ { 2 } + 42$

Divide. Simplify if possible. - $\frac { 3 y - 3 } { y } \div \frac { 5 y - 5 } { 8 y ^ { 2 } }$

Provide an appropriate response. -A formula for the focal length of a lens is $f = \frac { a b } { b + a }$ Calculate f (the focal length) for a = 8 cm and b = 17 cm. Round to the nearest tenth.

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { 3 x ^ { 2 } + 9 x ^ { 3 } } { 2 x + 6 x ^ { 2 } }$

Perform the indicated operation. Simplify if possible. - $\frac { 3 } { x + 13 } + \frac { 4 } { x + 13 }$

Perform the indicated operation. Simplify if possible. -A square shaped pasture has a side of length x 9+ 7 meters. Express the perimeter of the pasture as a rational expression.

Divide. Simplify if possible. - $\frac { x ^ { 2 } + 16 x + 63 } { x ^ { 2 } + 18 x + 81 } + \frac { x ^ { 2 } + 7 x } { x ^ { 2 } + 2 x - 63 }$

Perform the indicated operation. Simplify if possible. - $\frac { 3 y ^ { 2 } } { y - 1 } + \frac { - 3 y } { y - 1 }$

Simplify the rational expression. If the rational expression cannot be simplified, so state. - $\frac { y ^ { 2 } - 10 y + 24 } { 4 - y }$

Multiply. Simplify if possible. - $\frac { 2 p - 2 } { p } \cdot \frac { 7 p ^ { 2 } } { 6 p - 6 }$

Add or subtract as indicated. Simplify the result, if possible. - $\frac { 5 - x } { x - 6 } - \frac { 2 x + 6 } { 6 - x }$

Perform the indicated operation. Simplify if possible. - $\frac { 5 x + 4 } { 10 } + \frac { 7 x - 4 } { 10 }$

Perform the indicated operation(s). Simplify if possible. - $\frac { 6 } { 9 x ^ { 2 } } - \frac { 2 } { 9 x ^ { 2 } }$

Find the missing expression. - $\frac { 13 x - 3 } { x - 5 } - \frac { ? } { 5 - x } = \frac { 22 x + 1 } { x - 5 }$

Perform the indicated operation. Simplify if possible. - $\frac { x ^ { 2 } - 8 x } { x - 6 } + \frac { 12 } { x - 6 }$

Divide. Simplify if possible. - $\frac { x ^ { 2 } - 2 x + x y - 2 y } { 9 x ^ { 2 } - 9 y ^ { 2 } } \div \frac { x - 2 } { 12 x - 12 y }$

Multiply. Simplify if possible. - $\frac { k ^ { 2 } + 13 k + 36 } { k ^ { 2 } + 11 k + 18 } \cdot \frac { k ^ { 2 } + 4 k + 4 } { k ^ { 2 } + 6 k + 8 }$

Multiply. Simplify if possible. - $\frac { ( y - w ) ^ { 2 } } { y + w } \cdot \frac { y } { y ^ { 2 } - y w }$

Perform the indicated operation(s). Simplify if possible. - $\frac { 2 x + 7 } { x + 8 } - \frac { 3 x + 5 } { x + 8 } - \frac { 9 x } { x + 8 }$

Perform the indicated operation(s). Simplify if possible. - $\frac { 24 } { x - 8 } - \frac { 6 } { x - 8 }$

Find the least common denominator of the rational expressions. - $\frac { 6 } { 6 y + 42 }$ and $\frac { 3 } { y ^ { 2 } + 7 y }$

Variables, Real Numbers, and Mathematical Models

Linear Equations and Inequalities in One Variable

Linear Equations in Two Variables

Systems of Linear Equations

Exponents and Polynomials

Factoring Polynomials

Basics of Functions

Inequalities and Problem Solving

Radicals, Radical Functions, and Rational Exponents

Filters

Exam 7: Rational Expressions

Divide. Simplify if possible. - 3y−3y÷5y−58y2\frac { 3 y - 3 } { y } \div \frac { 5 y - 5 } { 8 y ^ { 2 } }y3y−3​÷8y25y−5​

Provide an appropriate response. -A formula for the focal length of a lens is f=abb+af = \frac { a b } { b + a }f=b+aab​ Calculate f (the focal length) for a = 8 cm and b = 17 cm. Round to the nearest tenth.

Simplify the rational expression. If the rational expression cannot be simplified, so state. - 3x2+9x32x+6x2\frac { 3 x ^ { 2 } + 9 x ^ { 3 } } { 2 x + 6 x ^ { 2 } }2x+6x23x2+9x3​

Perform the indicated operation. Simplify if possible. - 3x+13+4x+13\frac { 3 } { x + 13 } + \frac { 4 } { x + 13 }x+133​+x+134​

Perform the indicated operation. Simplify if possible. -A square shaped pasture has a side of length x 9+ 7 meters. Express the perimeter of the pasture as a rational expression.

Divide. Simplify if possible. - x2+16x+63x2+18x+81+x2+7xx2+2x−63\frac { x ^ { 2 } + 16 x + 63 } { x ^ { 2 } + 18 x + 81 } + \frac { x ^ { 2 } + 7 x } { x ^ { 2 } + 2 x - 63 }x2+18x+81x2+16x+63​+x2+2x−63x2+7x​

Perform the indicated operation. Simplify if possible. - 3y2y−1+−3yy−1\frac { 3 y ^ { 2 } } { y - 1 } + \frac { - 3 y } { y - 1 }y−13y2​+y−1−3y​

Simplify the rational expression. If the rational expression cannot be simplified, so state. - y2−10y+244−y\frac { y ^ { 2 } - 10 y + 24 } { 4 - y }4−yy2−10y+24​

Multiply. Simplify if possible. - 2p−2p⋅7p26p−6\frac { 2 p - 2 } { p } \cdot \frac { 7 p ^ { 2 } } { 6 p - 6 }p2p−2​⋅6p−67p2​

Add or subtract as indicated. Simplify the result, if possible. - 5−xx−6−2x+66−x\frac { 5 - x } { x - 6 } - \frac { 2 x + 6 } { 6 - x }x−65−x​−6−x2x+6​

Perform the indicated operation. Simplify if possible. - 5x+410+7x−410\frac { 5 x + 4 } { 10 } + \frac { 7 x - 4 } { 10 }105x+4​+107x−4​

Perform the indicated operation(s). Simplify if possible. - 69x2−29x2\frac { 6 } { 9 x ^ { 2 } } - \frac { 2 } { 9 x ^ { 2 } }9x26​−9x22​

Find the missing expression. - 13x−3x−5−?5−x=22x+1x−5\frac { 13 x - 3 } { x - 5 } - \frac { ? } { 5 - x } = \frac { 22 x + 1 } { x - 5 }x−513x−3​−5−x?​=x−522x+1​

Perform the indicated operation. Simplify if possible. - x2−8xx−6+12x−6\frac { x ^ { 2 } - 8 x } { x - 6 } + \frac { 12 } { x - 6 }x−6x2−8x​+x−612​

Divide. Simplify if possible. - x2−2x+xy−2y9x2−9y2÷x−212x−12y\frac { x ^ { 2 } - 2 x + x y - 2 y } { 9 x ^ { 2 } - 9 y ^ { 2 } } \div \frac { x - 2 } { 12 x - 12 y }9x2−9y2x2−2x+xy−2y​÷12x−12yx−2​

Multiply. Simplify if possible. - k2+13k+36k2+11k+18⋅k2+4k+4k2+6k+8\frac { k ^ { 2 } + 13 k + 36 } { k ^ { 2 } + 11 k + 18 } \cdot \frac { k ^ { 2 } + 4 k + 4 } { k ^ { 2 } + 6 k + 8 }k2+11k+18k2+13k+36​⋅k2+6k+8k2+4k+4​

Multiply. Simplify if possible. - (y−w)2y+w⋅yy2−yw\frac { ( y - w ) ^ { 2 } } { y + w } \cdot \frac { y } { y ^ { 2 } - y w }y+w(y−w)2​⋅y2−ywy​

Perform the indicated operation(s). Simplify if possible. - 2x+7x+8−3x+5x+8−9xx+8\frac { 2 x + 7 } { x + 8 } - \frac { 3 x + 5 } { x + 8 } - \frac { 9 x } { x + 8 }x+82x+7​−x+83x+5​−x+89x​

Perform the indicated operation(s). Simplify if possible. - 24x−8−6x−8\frac { 24 } { x - 8 } - \frac { 6 } { x - 8 }x−824​−x−86​

Find the least common denominator of the rational expressions. - 66y+42\frac { 6 } { 6 y + 42 }6y+426​ and 3y2+7y\frac { 3 } { y ^ { 2 } + 7 y }y2+7y3​