Question 1

Decide whether the statement is true or false. If false, correct the statement so it is true.
-$$| 6 - 12 | = | 6 | - | 12 |$$

Accepted Answer

A)  True 
B)  False; $| 6 - 12 | = | 12 | - | 6 |$ 
A)  True 
B)  False; $| 6 - 12 | = | 12 | - | 6 |$

Question 2

Solve the problem. Round to two decimal places unless otherwise indicated.
-Suppose the cost per ton, $y$, to build an oil platform of $x$ thousand tons is approximated by $y =$ $\frac { 212,500 } { x + 425 }$. What is the cost per ton for $x = 400 ?$

Accepted Answer

A)  $\$ 200,000.00$ 
B)  $\$ 257.58$ 
C)  $\$ 106.25$ 
D)  $\$ 103,030.30$ 
A)  $\$ 200,000.00$ 
B)  $\$ 257.58$ 
C)  $\$ 106.25$ 
D)  $\$ 103,030.30$

Question 3

Factor, using the given factor. Assume all variables represent positive real numbers.
-$x ^ { - 8 } - 6 x ^ { - 4 } ; \quad x ^ { - 8 }$

Accepted Answer

A)  $x ^ { - 8 } \left( 1 + 6 x ^ { 6 } \right)$ 
B)  $x ^ { - 8 } \left( x - 6 x ^ { 5 } \right)$ 
C)  $x ^ { - 8 } \left( 1 - 6 x ^ { 4 } \right)$ 
D)  $x ^ { - 8 } \left( x - 6 x ^ { 4 } \right)$ 
A)  $x ^ { - 8 } \left( 1 + 6 x ^ { 6 } \right)$ 
B)  $x ^ { - 8 } \left( x - 6 x ^ { 5 } \right)$ 
C)  $x ^ { - 8 } \left( 1 - 6 x ^ { 4 } \right)$ 
D)  $x ^ { - 8 } \left( x - 6 x ^ { 4 } \right)$

Question 4

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero
real numbers.
-$\frac { 5 ^ { - 8 } } { 5 ^ { - 5 } }$

Accepted Answer

A)  $\frac { 1 } { 513 }$ 
B)  $5 ^ { 3 }$ 
C)  $5 ^ { 13 }$ 
D)  $\frac { 1 } { 5 ^ { 3 } }$ 
A)  $\frac { 1 } { 513 }$ 
B)  $5 ^ { 3 }$ 
C)  $5 ^ { 13 }$ 
D)  $\frac { 1 } { 5 ^ { 3 } }$

Question 5

Simplify the expression. Assume all variables represent nonzero real numbers.
-$$\left( \frac { 2 x ^ { 3 } y ^ { 4 } } { z ^ { 4 } } \right) ^ { 3 }$$

Accepted Answer

A)  $\frac { 8 x ^ { 9 } y ^ { 12 } } { z ^ { 12 } }$ 
B)  $\frac { 2 x ^ { 9 } y ^ { 12 } } { z ^ { 12 } }$ 
C)  $\frac { 2 x ^ { 9 } y ^ { 12 } } { z ^ { 7 } }$ 
D)  $\frac { 8 x ^ { 6 } y ^ { 7 } } { z ^ { 7 } }$ 
A)  $\frac { 8 x ^ { 9 } y ^ { 12 } } { z ^ { 12 } }$ 
B)  $\frac { 2 x ^ { 9 } y ^ { 12 } } { z ^ { 12 } }$ 
C)  $\frac { 2 x ^ { 9 } y ^ { 12 } } { z ^ { 7 } }$ 
D)  $\frac { 8 x ^ { 6 } y ^ { 7 } } { z ^ { 7 } }$

Question 6

Simplify the expression. Assume all variables represent positive real numbers.
-$$7 \sqrt [ 4 ] { \mathrm { m } ^ { 9 } \mathrm { p } ^ { 6 } } - 2 \mathrm {~m} ^ { 2 } \mathrm { p } \sqrt [ 4 ] { \mathrm { mp } ^ { 2 } }$$

Accepted Answer

A)  $5 m ^ { 2 } p \sqrt [ 4 ] { m ^ { 2 } }$ 
B)  $9 m p ^ { 2 } \sqrt [ 4 ] { m p ^ { 2 } }$ 
C)  cannot be simplified 
D)  5 
A)  $5 m ^ { 2 } p \sqrt [ 4 ] { m ^ { 2 } }$ 
B)  $9 m p ^ { 2 } \sqrt [ 4 ] { m p ^ { 2 } }$ 
C)  cannot be simplified 
D)  5

Question 7

Find the product.
-$$( x + 12 ) ( 3 x - 11 )$$

Accepted Answer

A)  $3 x ^ { 2 } + 24 x - 132$ 
B)  $3 x ^ { 2 } - 132 x + 25$ 
C)  $x ^ { 2 } + 25 x + 25$ 
D)  $3 x ^ { 2 } + 25 x - 132$ 
A)  $3 x ^ { 2 } + 24 x - 132$ 
B)  $3 x ^ { 2 } - 132 x + 25$ 
C)  $x ^ { 2 } + 25 x + 25$ 
D)  $3 x ^ { 2 } + 25 x - 132$

Question 8

Calculate the value mentally.
-$\frac { 500^{1 / 2} } { 5^{1 / 2} }$

Accepted Answer

A)  100 
B)  10 
C)  1,000 
D)  none of these 
A)  100 
B)  10 
C)  1,000 
D)  none of these

Question 9

Factor by any method.
-$$9 a ^ { 3 } - 25 a$$

Accepted Answer

A)  $a ( 9 a + 1 ) ( a - 25 )$ 
B)  $a ( 3 a - 5 ) ^ { 2 }$ 
C)  $a ( 3 a + 5 ) ( 3 a - 5 )$ 
D)  $\left( 3 a ^ { 2 } + 5 \right) ( 3 a - 5 )$ 
A)  $a ( 9 a + 1 ) ( a - 25 )$ 
B)  $a ( 3 a - 5 ) ^ { 2 }$ 
C)  $a ( 3 a + 5 ) ( 3 a - 5 )$ 
D)  $\left( 3 a ^ { 2 } + 5 \right) ( 3 a - 5 )$

Question 10

Identify the expression as a polynomial or not a polynomial.
-$\frac { 2 } { x ^ { 3 } } + \frac { 5 } { x ^ { 2 } } - 10$

Accepted Answer

A)  Not a polynomial 
B)  Polynomial 
A)  Not a polynomial 
B)  Polynomial

Question 11

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not
zero.
-$$\frac { 5 \sqrt { x } } { \sqrt { x } - 5 \sqrt { y } }$$

Accepted Answer

A)  $\frac { 5 \sqrt { x } ( \sqrt { x } - 5 \sqrt { y } ) } { x + 25 y }$ 
B)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x - 5 y }$ 
C)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x - 25 y }$ 
D)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x + 5 y }$ 
A)  $\frac { 5 \sqrt { x } ( \sqrt { x } - 5 \sqrt { y } ) } { x + 25 y }$ 
B)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x - 5 y }$ 
C)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x - 25 y }$ 
D)  $\frac { 5 \sqrt { x } ( \sqrt { x } + 5 \sqrt { y } ) } { x + 5 y }$

Question 12

Find the sum or difference.
-$$\left( - 2 x ^ { 2 } + 6 x ^ { 4 } + 3 - 4 x ^ { 3 } \right) - \left( - 9 - 7 x ^ { 3 } + 4 x ^ { 4 } + 5 x ^ { 2 } \right)$$

Accepted Answer

A)  $2 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } - 6$ 
B)  $10 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } - 6$ 
C)  $10 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } + 12$ 
D)  $2 x ^ { 4 } + 3 x ^ { 3 } - 7 x ^ { 2 } + 12$ 
A)  $2 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } - 6$ 
B)  $10 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } - 6$ 
C)  $10 x ^ { 4 } - 11 x ^ { 3 } + 3 x ^ { 2 } + 12$ 
D)  $2 x ^ { 4 } + 3 x ^ { 3 } - 7 x ^ { 2 } + 12$

Question 13

Write in exponential form. Assume all variables are positive real numbers.
-$$- \sqrt { 294 \mathrm {~m} ^ { 11 } }$$

Accepted Answer

A)  $- 86,436 \mathrm {~m} ^{ { 2/11 }}$ 
B)  $- 294^{1 / 2} \mathrm {~m}^{ 11 / 2}$ 
C)  $294 \mathrm {~m}^{ 2 / 11}$ 
D)  $- 294 \mathrm {~m} ^ { 11 / 2 }$ 
A)  $- 86,436 \mathrm {~m} ^{ { 2/11 }}$ 
B)  $- 294^{1 / 2} \mathrm {~m}^{ 11 / 2}$ 
C)  $294 \mathrm {~m}^{ 2 / 11}$ 
D)  $- 294 \mathrm {~m} ^ { 11 / 2 }$

Question 14

Tell whether the statement is true or false.
-Let $\mathrm { A } = \{ 9,10,11,12,13,14 \} , \mathrm { B } = \{ 11,13,15 \} , \mathrm { C } = \{ 9,10,12,14 \} , \mathrm { D } = \{ 9,14 \}$, and $U = \{ 9,10,11,12,13,14,15 \}$.
$C \subseteq \varnothing$

Accepted Answer

A)  True 
B)  False 
A)  True 
B)  False

Question 15

Identify the expression as a polynomial or not a polynomial.
-$2 x - \frac { 3 } { x } + 12$

Accepted Answer

A)  Not a polynomial 
B)  Polynomial 
A)  Not a polynomial 
B)  Polynomial

Question 16

Find the product or quotient.
-$$\frac { ( x - y ) ^ { 2 } - 16 } { ( x + y ) ^ { 2 } - 16 } \div \frac { x - y - 4 } { x + y + 4 }$$

Accepted Answer

A)  -1 
B)  $\frac { x - y + 4 } { x + y - 4 }$ 
C)  $x - y + 4$ 
D)  $\frac { x + y + 4 } { x - y - 4 }$ 
A)  -1 
B)  $\frac { x - y + 4 } { x + y - 4 }$ 
C)  $x - y + 4$ 
D)  $\frac { x + y + 4 } { x - y - 4 }$

Question 17

Perform the indicated operation and write your answer with positive integer exponents.
-$$\frac { m ^ { - 1 } + n ^ { - 1 } } { ( m n ) ^ { - 1 } }$$

Accepted Answer

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

Question 18

Factor the trinomial, if possible.
-$$x ^ { 2 } - x - 63$$

Accepted Answer

A)  $( x - 7 ) ( x + 9 )$ 
B)  $( x - 63 ) ( x + 1 )$ 
C)  $( x + 7 ) ( x - 9 )$ 
D)  prime 
A)  $( x - 7 ) ( x + 9 )$ 
B)  $( x - 63 ) ( x + 1 )$ 
C)  $( x + 7 ) ( x - 9 )$ 
D)  prime

Question 19

Simplify the expression. Assume all variables represent nonzero real numbers.
-$\left( 5 ^ { 3 } \right) ^ { 3 }$

Accepted Answer

A)   $5^{6}$ 
B)  $25 ^ { 9 }$ 
C)  $5 ^ { 9 }$ 
D)  $25^{6}$ 
A)   $5^{6}$ 
B)  $25 ^ { 9 }$ 
C)  $5 ^ { 9 }$ 
D)  $25^{6}$

Question 20

Use set notation, and list all the elements of the set.
-$$\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \ldots , \frac { 1 } { 243 } \right\}$$

Accepted Answer

A)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 729 } , \frac { 1 } { 19,683 } \right\}$ 
B)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 243 } \right\}$ 
C)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 81 } , \frac { 1 } { 243 } \right\}$ 
D)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 5 } , \frac { 1 } { 6 } , \frac { 1 } { 7 } , \frac { 1 } { 8 } \right\}$ 
A)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 729 } , \frac { 1 } { 19,683 } \right\}$ 
B)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 243 } \right\}$ 
C)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \frac { 1 } { 27 } , \frac { 1 } { 81 } , \frac { 1 } { 243 } \right\}$ 
D)  $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 5 } , \frac { 1 } { 6 } , \frac { 1 } { 7 } , \frac { 1 } { 8 } \right\}$

Decide whether the statement is true or false. If false, correct the statement so it is true. - $| 6 - 12 | = | 6 | - | 12 |$

Solve the problem. Round to two decimal places unless otherwise indicated. -Suppose the cost per ton, $y$ , to build an oil platform of $x$ thousand tons is approximated by $y =$ $\frac { 212,500 } { x + 425 }$ . What is the cost per ton for $x = 400 ?$

Factor, using the given factor. Assume all variables represent positive real numbers. - $x ^ { - 8 } - 6 x ^ { - 4 } ; \quad x ^ { - 8 }$

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero real numbers. - $\frac { 5 ^ { - 8 } } { 5 ^ { - 5 } }$

Simplify the expression. Assume all variables represent nonzero real numbers. - $\left( \frac { 2 x ^ { 3 } y ^ { 4 } } { z ^ { 4 } } \right) ^ { 3 }$

Simplify the expression. Assume all variables represent positive real numbers. - $7 \sqrt [ 4 ] { \mathrm { m } ^ { 9 } \mathrm { p } ^ { 6 } } - 2 \mathrm {~m} ^ { 2 } \mathrm { p } \sqrt [ 4 ] { \mathrm { mp } ^ { 2 } }$

Find the product. - $( x + 12 ) ( 3 x - 11 )$

Calculate the value mentally. - $\frac { 500^{1 / 2} } { 5^{1 / 2} }$

Factor by any method. - $9 a ^ { 3 } - 25 a$

Identify the expression as a polynomial or not a polynomial. - $\frac { 2 } { x ^ { 3 } } + \frac { 5 } { x ^ { 2 } } - 10$

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not zero. - $\frac { 5 \sqrt { x } } { \sqrt { x } - 5 \sqrt { y } }$

Find the sum or difference. - $\left( - 2 x ^ { 2 } + 6 x ^ { 4 } + 3 - 4 x ^ { 3 } \right) - \left( - 9 - 7 x ^ { 3 } + 4 x ^ { 4 } + 5 x ^ { 2 } \right)$

Write in exponential form. Assume all variables are positive real numbers. - $- \sqrt { 294 \mathrm {~m} ^ { 11 } }$

Tell whether the statement is true or false. -Let $\mathrm { A } = \{ 9,10,11,12,13,14 \} , \mathrm { B } = \{ 11,13,15 \} , \mathrm { C } = \{ 9,10,12,14 \} , \mathrm { D } = \{ 9,14 \}$ , and $U = \{ 9,10,11,12,13,14,15 \}$ . $C \subseteq \varnothing$

Identify the expression as a polynomial or not a polynomial. - $2 x - \frac { 3 } { x } + 12$

Find the product or quotient. - $\frac { ( x - y ) ^ { 2 } - 16 } { ( x + y ) ^ { 2 } - 16 } \div \frac { x - y - 4 } { x + y + 4 }$

Perform the indicated operation and write your answer with positive integer exponents. - $\frac { m ^ { - 1 } + n ^ { - 1 } } { ( m n ) ^ { - 1 } }$

Factor the trinomial, if possible. - $x ^ { 2 } - x - 63$

Simplify the expression. Assume all variables represent nonzero real numbers. - $\left( 5 ^ { 3 } \right) ^ { 3 }$

Use set notation, and list all the elements of the set. - $\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \ldots , \frac { 1 } { 243 } \right\}$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 12: Review of Basic Concepts

Decide whether the statement is true or false. If false, correct the statement so it is true. - ∣6−12∣=∣6∣−∣12∣| 6 - 12 | = | 6 | - | 12 |∣6−12∣=∣6∣−∣12∣

Solve the problem. Round to two decimal places unless otherwise indicated. -Suppose the cost per ton, yyy , to build an oil platform of xxx thousand tons is approximated by y=y =y= 212,500x+425\frac { 212,500 } { x + 425 }x+425212,500​ . What is the cost per ton for x=400?x = 400 ?x=400?

Factor, using the given factor. Assume all variables represent positive real numbers. - x−8−6x−4;x−8x ^ { - 8 } - 6 x ^ { - 4 } ; \quad x ^ { - 8 }x−8−6x−4;x−8

Perform the indicated operations. Write the result using only positive exponents. Assume all variables represent nonzero real numbers. - 5−85−5\frac { 5 ^ { - 8 } } { 5 ^ { - 5 } }5−55−8​

Simplify the expression. Assume all variables represent nonzero real numbers. - (2x3y4z4)3\left( \frac { 2 x ^ { 3 } y ^ { 4 } } { z ^ { 4 } } \right) ^ { 3 }(z42x3y4​)3

Simplify the expression. Assume all variables represent positive real numbers. - 7m9p64−2 m2pmp247 \sqrt [ 4 ] { \mathrm { m } ^ { 9 } \mathrm { p } ^ { 6 } } - 2 \mathrm {~m} ^ { 2 } \mathrm { p } \sqrt [ 4 ] { \mathrm { mp } ^ { 2 } }74m9p6​−2 m2p4mp2​

Find the product. - (x+12)(3x−11)( x + 12 ) ( 3 x - 11 )(x+12)(3x−11)

Calculate the value mentally. - 5001/251/2\frac { 500^{1 / 2} } { 5^{1 / 2} }51/25001/2​

Factor by any method. - 9a3−25a9 a ^ { 3 } - 25 a9a3−25a

Identify the expression as a polynomial or not a polynomial. - 2x3+5x2−10\frac { 2 } { x ^ { 3 } } + \frac { 5 } { x ^ { 2 } } - 10x32​+x25​−10

Rationalize the denominator. Assume that all variables represent nonnegative numbers and that the denominator is not zero. - 5xx−5y\frac { 5 \sqrt { x } } { \sqrt { x } - 5 \sqrt { y } }x​−5y​5x​​

Find the sum or difference. - (−2x2+6x4+3−4x3)−(−9−7x3+4x4+5x2)\left( - 2 x ^ { 2 } + 6 x ^ { 4 } + 3 - 4 x ^ { 3 } \right) - \left( - 9 - 7 x ^ { 3 } + 4 x ^ { 4 } + 5 x ^ { 2 } \right)(−2x2+6x4+3−4x3)−(−9−7x3+4x4+5x2)

Write in exponential form. Assume all variables are positive real numbers. - −294 m11- \sqrt { 294 \mathrm {~m} ^ { 11 } }−294 m11​

Identify the expression as a polynomial or not a polynomial. - 2x−3x+122 x - \frac { 3 } { x } + 122x−x3​+12

Find the product or quotient. - (x−y)2−16(x+y)2−16÷x−y−4x+y+4\frac { ( x - y ) ^ { 2 } - 16 } { ( x + y ) ^ { 2 } - 16 } \div \frac { x - y - 4 } { x + y + 4 }(x+y)2−16(x−y)2−16​÷x+y+4x−y−4​

Perform the indicated operation and write your answer with positive integer exponents. - m−1+n−1(mn)−1\frac { m ^ { - 1 } + n ^ { - 1 } } { ( m n ) ^ { - 1 } }(mn)−1m−1+n−1​

Factor the trinomial, if possible. - x2−x−63x ^ { 2 } - x - 63x2−x−63

Simplify the expression. Assume all variables represent nonzero real numbers. - (53)3\left( 5 ^ { 3 } \right) ^ { 3 }(53)3

Use set notation, and list all the elements of the set. - {1,13,19,…,1243}\left\{ 1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \ldots , \frac { 1 } { 243 } \right\}{1,31​,91​,…,2431​}