Question 1

Perform the division. $\frac { 16 a ^ { 7 } b ^ { 6 } - 24 a ^ { 6 } b ^ { 7 } } { 4 a ^ { 6 } b ^ { 6 } }$

Accepted Answer

A)  $4 b - 6 a$ 
B)  $2 a - 2 b$ 
C)  $4 a - 6 b$ 
D)  $4 b - 7 a$ 
E)  $2 a ^ { 2 } - 2 b$ 
A)  $4 b - 6 a$ 
B)  $2 a - 2 b$ 
C)  $4 a - 6 b$ 
D)  $4 b - 7 a$ 
E)  $2 a ^ { 2 } - 2 b$

Question 2

Find the quotient. $\frac { 1 } { 2 } \div \frac { 5 } { 3 }$

Accepted Answer

A)  $\frac { 2 } { 7 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 7 } { 13 }$ 
E)  $\frac { 8 } { 11 }$ 
A)  $\frac { 2 } { 7 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 7 } { 13 }$ 
E)  $\frac { 8 } { 11 }$

Question 3

Use synthetic division to perform the division. $\left( x ^ { 2 } - 5 x + 14 \right) \div ( x + 3 )$

Accepted Answer

The answer of Use synthetic division to perform the division....

Question 4

Use a calculator and synthetic division to perform the division. $\left( 15 x ^ { 3 } - 25 \right) \div ( x + 53 )$

Accepted Answer

A)  $15 x ^ { 2 } - 795 x + 42,135 - \frac { 2,233,180 } { x + 53 }$ 
B)  $15 x ^ { 2 } - 795 x + 42,180 - \frac { 2,233,250 } { x + 53 }$ 
C)  $15 x ^ { 2 } - 795 x + 42,160 - \frac { 2,233,250 } { x + 53 }$ 
D)  $15 x ^ { 2 } - 795 x + 42,170 - \frac { 2,233,240 } { x + 53 }$ 
A)  $15 x ^ { 2 } - 795 x + 42,135 - \frac { 2,233,180 } { x + 53 }$ 
B)  $15 x ^ { 2 } - 795 x + 42,180 - \frac { 2,233,250 } { x + 53 }$ 
C)  $15 x ^ { 2 } - 795 x + 42,160 - \frac { 2,233,250 } { x + 53 }$ 
D)  $15 x ^ { 2 } - 795 x + 42,170 - \frac { 2,233,240 } { x + 53 }$

Question 5

Perform the operation involving rational numbers. Be sure to express your answer in reduced form. $\frac { 1 } { 3 } + \frac { 5 } { 6 } - \frac { 4 } { 9 }$

Accepted Answer

A)  $\frac { 29 } { 18 }$ 
B)  $\frac { 13 } { 9 }$ 
C)  $\frac { 13 } { 17 }$ 
D)  $\frac { 13 } { 18 }$ 
E)  $\frac { 18 } { 13 }$ 
A)  $\frac { 29 } { 18 }$ 
B)  $\frac { 13 } { 9 }$ 
C)  $\frac { 13 } { 17 }$ 
D)  $\frac { 13 } { 18 }$ 
E)  $\frac { 18 } { 13 }$

Question 6

An inlet pipe can fill a tank in 8 minutes. A drain can empty the tank in 10 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

Accepted Answer

A) 38 minutes 
B) 20 minutes 
C) 2 minutes 
D) 50 minutes 
E) 40 minutes 
A) 38 minutes 
B) 20 minutes 
C) 2 minutes 
D) 50 minutes 
E) 40 minutes

Question 7

To travel 68 miles, it takes Sue, riding a moped, 4 hours less time than it takes Doreen to travel 80 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

Accepted Answer

A) Sue: 17 mph for 4 hours, Doreen: 10 mph for 8 hours 
B) Sue: 10 mph for 4 hours, Doreen: 17 mph for 8 hours 
C) Sue: 21 mph for 8 hours, Doreen: 14 mph for 12 hours 
D) Sue: 4 mph for 17 hours, Doreen: 10 mph for 8 hours 
E) Sue: 27 mph for 4 hours, Doreen: 20 mph for 8 hours 
A) Sue: 17 mph for 4 hours, Doreen: 10 mph for 8 hours 
B) Sue: 10 mph for 4 hours, Doreen: 17 mph for 8 hours 
C) Sue: 21 mph for 8 hours, Doreen: 14 mph for 12 hours 
D) Sue: 4 mph for 17 hours, Doreen: 10 mph for 8 hours 
E) Sue: 27 mph for 4 hours, Doreen: 20 mph for 8 hours

Question 8

Perform the multiplication. Simplify the result when possible. $\frac { 5 } { 63 } \cdot \frac { 72 } { 5 }$

Accepted Answer

A)  $\frac { 10 } { 9 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 10 } { 11 }$ 
D)  $\frac { 11 } { 10 }$ 
E)  9
$$8$$ 
A)  $\frac { 10 } { 9 }$ 
B)  $\frac { 8 } { 7 }$ 
C)  $\frac { 10 } { 11 }$ 
D)  $\frac { 11 } { 10 }$ 
E)  9
$$8$$

Question 9

Solve the equation. $\frac { y } { 4 } + \frac { y - 68 } { 4 } = \frac { 3 y + 6 } { 2 }$

Accepted Answer

A) y = -4 
B) y = -5 
C) y = -19 
D) y = -22 
E) y = -20 
A) y = -4 
B) y = -5 
C) y = -19 
D) y = -22 
E) y = -20

Question 10

Perform the operation. $\frac { 1 } { 2 } - \left( - \frac { 3 } { 5 } \right)$

Accepted Answer

A)  $\frac { 11 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  $- \frac { 1 } { 10 }$ 
A)  $\frac { 11 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 10 }$ 
D)  $\frac { 1 } { 5 }$ 
E)  $- \frac { 1 } { 10 }$

Question 11

Add or subtract the rational expressions. Be sure to express your answer in simplest form. $\frac { 5 } { 4 x } + \frac { 6 } { 5 y } - 2$

Accepted Answer

A)  $\frac { 20 y + 30 x - 8 x - 10 y } { 4 x + 5 y }$ 
B)  $\frac { 25 y + 24 x - 40 x y } { 20 x y }$ 
C)  $\frac { 25 y + 24 x } { 4 x + 5 y } - 2$ 
D)  $\frac { 25 y + 24 x } { 20 x y }$ 
E)  $\frac { 25 y + 24 x - x y } { 20 x y }$ 
A)  $\frac { 20 y + 30 x - 8 x - 10 y } { 4 x + 5 y }$ 
B)  $\frac { 25 y + 24 x - 40 x y } { 20 x y }$ 
C)  $\frac { 25 y + 24 x } { 4 x + 5 y } - 2$ 
D)  $\frac { 25 y + 24 x } { 20 x y }$ 
E)  $\frac { 25 y + 24 x - x y } { 20 x y }$

Question 12

Solve the equation. $\frac { 5 n } { 6 } - \frac { n } { 8 } = \frac { - 102 } { 24 }$

Accepted Answer

A) n = -6 
B) n = -12 
C) n = 6 
D) n = 8 
E) n = -9 
A) n = -6 
B) n = -12 
C) n = 6 
D) n = 8 
E) n = -9

Question 13

Simplify the complex fraction. $\frac { \frac { 1 } { 2 } - \frac { 1 } { 4 } } { \frac { 5 } { 6 } + \frac { 3 } { 4 } }$

Accepted Answer

A)  $\frac { 1 } { 21 }$ 
B)  $\frac { 3 } { 21 }$ 
C)  $\frac { 1 } { 19 }$ 
D)  $\frac { 2 } { 19 }$ 
E)  $\frac { 3 } { 19 }$ 
A)  $\frac { 1 } { 21 }$ 
B)  $\frac { 3 } { 21 }$ 
C)  $\frac { 1 } { 19 }$ 
D)  $\frac { 2 } { 19 }$ 
E)  $\frac { 3 } { 19 }$

Question 14

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. $\frac { 4 } { 9 } \cdot \frac { 6 } { 13 } \div \frac { 2 } { 9 }$

Accepted Answer

A)  $1$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 11 } { 12 }$ 
D)  $\frac { 1 } { 12 }$ 
E)  $\frac { 12 } { 13 }$ 
A)  $1$ 
B)  $\frac { 1 } { 13 }$ 
C)  $\frac { 11 } { 12 }$ 
D)  $\frac { 1 } { 12 }$ 
E)  $\frac { 12 } { 13 }$

Question 15

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $\frac { 9 y + 9 a } { y ^ { 2 } + y a }$

Accepted Answer

A)  $\frac { 9 } { a }$ 
B)  $\frac { 9 } { y + a }$ 
C)  $\frac { 9 } { y }$ 
D)  $\frac { 9 y } { a }$ 
E) in lowest terms 
A)  $\frac { 9 } { a }$ 
B)  $\frac { 9 } { y + a }$ 
C)  $\frac { 9 } { y }$ 
D)  $\frac { 9 y } { a }$ 
E) in lowest terms

Question 16

Add the rational expressions. Be sure to express your answer in simplest form. $\frac { x - 5 } { 6 } + \frac { x + 7 } { 7 }$

Accepted Answer

A)  $\frac { 36 } { 13 x + 7 }$ 
B)  $\frac { 13 x - 77 } { 42 }$ 
C)  $\frac { 13 x + 7 } { 42 }$ 
D)  $\frac { 13 x - 7 } { 42 }$ 
E)  $\frac { 2 x + 7 } { 13 }$ 
A)  $\frac { 36 } { 13 x + 7 }$ 
B)  $\frac { 13 x - 77 } { 42 }$ 
C)  $\frac { 13 x + 7 } { 42 }$ 
D)  $\frac { 13 x - 7 } { 42 }$ 
E)  $\frac { 2 x + 7 } { 13 }$

Question 17

Perform the indicated operation and express your answer in simplest form. $\frac { 5 x } { x ^ { 2 } + 8 x } + \frac { 6 } { x }$

Accepted Answer

A)  $\frac { x + 48 } { x + 8 }$ 
B)  $\frac { 5 x + 48 } { x ( x + 8 ) }$ 
C)  $\frac { 11 x + 48 } { x + 8 }$ 
D)  $\frac { 11 x + 48 } { x + 2 }$ 
E)  $\frac { 11 x + 48 } { x ( x + 8 ) }$ 
A)  $\frac { x + 48 } { x + 8 }$ 
B)  $\frac { 5 x + 48 } { x ( x + 8 ) }$ 
C)  $\frac { 11 x + 48 } { x + 8 }$ 
D)  $\frac { 11 x + 48 } { x + 2 }$ 
E)  $\frac { 11 x + 48 } { x ( x + 8 ) }$

Question 18

Use synthetic division to perform the division. $\left( x ^ { 2 } - 12 x + 32 \right) \div ( x - 4 )$

Accepted Answer

A) x + 10 
B) x + 7 
C) x - 7 
D) x - 8 
E) x - 10 
A) x + 10 
B) x + 7 
C) x - 7 
D) x - 8 
E) x - 10

Question 19

Simplify the complex fraction. $\frac { \frac { 5 } { x } - \frac { 6 } { y } } { \frac { 7 } { y } - \frac { 2 } { x y } }$

Accepted Answer

A)  $\frac { x - 6 y } { 7 x - 2 }$ 
B)  $\frac { 5 y - 6 x } { 7 x - 2 }$ 
C)  $\frac { x - 7 y } { x y ^ { 2 } }$ 
D)  $\frac { x - 7 y } { x y }$ 
E)  $\frac { 5 x - 6 y } { 7 - 2 x }$ 
A)  $\frac { x - 6 y } { 7 x - 2 }$ 
B)  $\frac { 5 y - 6 x } { 7 x - 2 }$ 
C)  $\frac { x - 7 y } { x y ^ { 2 } }$ 
D)  $\frac { x - 7 y } { x y }$ 
E)  $\frac { 5 x - 6 y } { 7 - 2 x }$

Question 20

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Accepted Answer

A) h = 27 ft 
B) h = 42 ft 
C) h = 30 ft 
D) h = 35 ft 
E) h = 22 ft 
A) h = 27 ft 
B) h = 42 ft 
C) h = 30 ft 
D) h = 35 ft 
E) h = 22 ft

Perform the division. $\frac { 16 a ^ { 7 } b ^ { 6 } - 24 a ^ { 6 } b ^ { 7 } } { 4 a ^ { 6 } b ^ { 6 } }$

Find the quotient. $\frac { 1 } { 2 } \div \frac { 5 } { 3 }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 5 x + 14 \right) \div ( x + 3 )$

Use a calculator and synthetic division to perform the division. $\left( 15 x ^ { 3 } - 25 \right) \div ( x + 53 )$

Perform the operation involving rational numbers. Be sure to express your answer in reduced form. $\frac { 1 } { 3 } + \frac { 5 } { 6 } - \frac { 4 } { 9 }$

An inlet pipe can fill a tank in 8 minutes. A drain can empty the tank in 10 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

To travel 68 miles, it takes Sue, riding a moped, 4 hours less time than it takes Doreen to travel 80 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

Perform the multiplication. Simplify the result when possible. $\frac { 5 } { 63 } \cdot \frac { 72 } { 5 }$

Solve the equation. $\frac { y } { 4 } + \frac { y - 68 } { 4 } = \frac { 3 y + 6 } { 2 }$

Perform the operation. $\frac { 1 } { 2 } - \left( - \frac { 3 } { 5 } \right)$

Add or subtract the rational expressions. Be sure to express your answer in simplest form. $\frac { 5 } { 4 x } + \frac { 6 } { 5 y } - 2$

Solve the equation. $\frac { 5 n } { 6 } - \frac { n } { 8 } = \frac { - 102 } { 24 }$

Simplify the complex fraction. $\frac { \frac { 1 } { 2 } - \frac { 1 } { 4 } } { \frac { 5 } { 6 } + \frac { 3 } { 4 } }$

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. $\frac { 4 } { 9 } \cdot \frac { 6 } { 13 } \div \frac { 2 } { 9 }$

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $\frac { 9 y + 9 a } { y ^ { 2 } + y a }$

Add the rational expressions. Be sure to express your answer in simplest form. $\frac { x - 5 } { 6 } + \frac { x + 7 } { 7 }$

Perform the indicated operation and express your answer in simplest form. $\frac { 5 x } { x ^ { 2 } + 8 x } + \frac { 6 } { x }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 12 x + 32 \right) \div ( x - 4 )$

Simplify the complex fraction. $\frac { \frac { 5 } { x } - \frac { 6 } { y } } { \frac { 7 } { y } - \frac { 2 } { x y } }$

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Basic Concepts and Properties

Equations, Inequalities, and Problem Solving

Polynomials

Exponents and Radicals

Quadratic Equations and Inequalities

Equations and Inequalities in Two Variables

Conic Sections

Functions

Systems of Equations

Exponential and Logarithmic Functions

Filters

Exam 4: Rational Expressions

Perform the division. 16a7b6−24a6b74a6b6\frac { 16 a ^ { 7 } b ^ { 6 } - 24 a ^ { 6 } b ^ { 7 } } { 4 a ^ { 6 } b ^ { 6 } }4a6b616a7b6−24a6b7​

Find the quotient. 12÷53\frac { 1 } { 2 } \div \frac { 5 } { 3 }21​÷35​

Use synthetic division to perform the division. (x2−5x+14)÷(x+3)\left( x ^ { 2 } - 5 x + 14 \right) \div ( x + 3 )(x2−5x+14)÷(x+3)

Use a calculator and synthetic division to perform the division. (15x3−25)÷(x+53)\left( 15 x ^ { 3 } - 25 \right) \div ( x + 53 )(15x3−25)÷(x+53)

Perform the operation involving rational numbers. Be sure to express your answer in reduced form. 13+56−49\frac { 1 } { 3 } + \frac { 5 } { 6 } - \frac { 4 } { 9 }31​+65​−94​

An inlet pipe can fill a tank in 8 minutes. A drain can empty the tank in 10 minutes. If the tank is empty and both the pipe and drain are open, how long will it take before the tank overflows?

To travel 68 miles, it takes Sue, riding a moped, 4 hours less time than it takes Doreen to travel 80 miles riding a bicycle. Sue travels 7 miles per hour faster than Doreen. Find the times and rates of both girls.

Perform the multiplication. Simplify the result when possible. 563⋅725\frac { 5 } { 63 } \cdot \frac { 72 } { 5 }635​⋅572​

Solve the equation. y4+y−684=3y+62\frac { y } { 4 } + \frac { y - 68 } { 4 } = \frac { 3 y + 6 } { 2 }4y​+4y−68​=23y+6​

Perform the operation. 12−(−35)\frac { 1 } { 2 } - \left( - \frac { 3 } { 5 } \right)21​−(−53​)

Add or subtract the rational expressions. Be sure to express your answer in simplest form. 54x+65y−2\frac { 5 } { 4 x } + \frac { 6 } { 5 y } - 24x5​+5y6​−2

Solve the equation. 5n6−n8=−10224\frac { 5 n } { 6 } - \frac { n } { 8 } = \frac { - 102 } { 24 }65n​−8n​=24−102​

Simplify the complex fraction. 12−1456+34\frac { \frac { 1 } { 2 } - \frac { 1 } { 4 } } { \frac { 5 } { 6 } + \frac { 3 } { 4 } }65​+43​21​−41​​

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. 49⋅613÷29\frac { 4 } { 9 } \cdot \frac { 6 } { 13 } \div \frac { 2 } { 9 }94​⋅136​÷92​

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. 9y+9ay2+ya\frac { 9 y + 9 a } { y ^ { 2 } + y a }y2+ya9y+9a​

Add the rational expressions. Be sure to express your answer in simplest form. x−56+x+77\frac { x - 5 } { 6 } + \frac { x + 7 } { 7 }6x−5​+7x+7​

Perform the indicated operation and express your answer in simplest form. 5xx2+8x+6x\frac { 5 x } { x ^ { 2 } + 8 x } + \frac { 6 } { x }x2+8x5x​+x6​

Use synthetic division to perform the division. (x2−12x+32)÷(x−4)\left( x ^ { 2 } - 12 x + 32 \right) \div ( x - 4 )(x2−12x+32)÷(x−4)

Simplify the complex fraction. 5x−6y7y−2xy\frac { \frac { 5 } { x } - \frac { 6 } { y } } { \frac { 7 } { y } - \frac { 2 } { x y } }y7​−xy2​x5​−y6​​

Use similar triangles to help solve the problem. A tree casts a shadow of 24 feet at the same time as a 5-foot tall man casts a shadow of 4 feet. Find the height of the tree.

Basic Concepts and Properties

Equations, Inequalities, and Problem Solving

Polynomials

Exponents and Radicals

Quadratic Equations and Inequalities

Equations and Inequalities in Two Variables

Conic Sections

Functions

Systems of Equations

Exponential and Logarithmic Functions

Filters

Perform the division. $\frac { 16 a ^ { 7 } b ^ { 6 } - 24 a ^ { 6 } b ^ { 7 } } { 4 a ^ { 6 } b ^ { 6 } }$

Find the quotient. $\frac { 1 } { 2 } \div \frac { 5 } { 3 }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 5 x + 14 \right) \div ( x + 3 )$

Use a calculator and synthetic division to perform the division. $\left( 15 x ^ { 3 } - 25 \right) \div ( x + 53 )$

Perform the operation involving rational numbers. Be sure to express your answer in reduced form. $\frac { 1 } { 3 } + \frac { 5 } { 6 } - \frac { 4 } { 9 }$

Perform the multiplication. Simplify the result when possible. $\frac { 5 } { 63 } \cdot \frac { 72 } { 5 }$

Solve the equation. $\frac { y } { 4 } + \frac { y - 68 } { 4 } = \frac { 3 y + 6 } { 2 }$

Perform the operation. $\frac { 1 } { 2 } - \left( - \frac { 3 } { 5 } \right)$

Add or subtract the rational expressions. Be sure to express your answer in simplest form. $\frac { 5 } { 4 x } + \frac { 6 } { 5 y } - 2$

Solve the equation. $\frac { 5 n } { 6 } - \frac { n } { 8 } = \frac { - 102 } { 24 }$

Simplify the complex fraction. $\frac { \frac { 1 } { 2 } - \frac { 1 } { 4 } } { \frac { 5 } { 6 } + \frac { 3 } { 4 } }$

Perform the indicated operations involving rational numbers. Express the final answer in reduced form. $\frac { 4 } { 9 } \cdot \frac { 6 } { 13 } \div \frac { 2 } { 9 }$

Write the expression in lowest terms. If the expression is already in lowest terms, so indicate. Assume that the denominator does not equal zero. $\frac { 9 y + 9 a } { y ^ { 2 } + y a }$

Add the rational expressions. Be sure to express your answer in simplest form. $\frac { x - 5 } { 6 } + \frac { x + 7 } { 7 }$

Perform the indicated operation and express your answer in simplest form. $\frac { 5 x } { x ^ { 2 } + 8 x } + \frac { 6 } { x }$

Use synthetic division to perform the division. $\left( x ^ { 2 } - 12 x + 32 \right) \div ( x - 4 )$

Simplify the complex fraction. $\frac { \frac { 5 } { x } - \frac { 6 } { y } } { \frac { 7 } { y } - \frac { 2 } { x y } }$