Question 1

Add or subtract. Assume all variables represent positive real numbers.
-$\sqrt { 5 a } - 2 \sqrt { 80 a } - 6 \sqrt { 125 a }$

Accepted Answer

A)  $- 8 \sqrt { 210 a }$ 
B)  $- 37 \sqrt { 5 a }$ 
C)  $- 8 \sqrt { 5 a }$ 
D)  $- 37 \sqrt { 210 a }$ 
A)  $- 8 \sqrt { 210 a }$ 
B)  $- 37 \sqrt { 5 a }$ 
C)  $- 8 \sqrt { 5 a }$ 
D)  $- 37 \sqrt { 210 a }$

Question 2

Use the quotient rule to divide and simplify.
-$\sqrt { \frac { 486 \mathrm { r } ^ { 2 } \mathrm { y } } { \mathrm { x } ^ { 4 } } }$

Accepted Answer

A)  $\frac { 9 r \sqrt { 6 y } } { x ^ { 2 } }$ 
B)  $\frac { r \sqrt { 486 y } } { x ^ { 2 } }$ 
C)  $\frac { 9 \sqrt { 6 r ^ { 2 } y } } { x ^ { 2 } }$ 
D)  $\frac { 81 r \sqrt { 6 y } } { x ^ { 2 } }$ 
A)  $\frac { 9 r \sqrt { 6 y } } { x ^ { 2 } }$ 
B)  $\frac { r \sqrt { 486 y } } { x ^ { 2 } }$ 
C)  $\frac { 9 \sqrt { 6 r ^ { 2 } y } } { x ^ { 2 } }$ 
D)  $\frac { 81 r \sqrt { 6 y } } { x ^ { 2 } }$

Question 3

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 30 } { \sqrt { 6 x } }$

Accepted Answer

A)  $\frac { 5 \sqrt { 6 x } } { 6 x }$ 
B)  $\frac { 5 \sqrt { 6 x } } { x }$ 
C)  $\frac { 30 \sqrt { x } } { x }$ 
D)  $\frac { 30 \sqrt { 6 x } } { 6 x }$ 
A)  $\frac { 5 \sqrt { 6 x } } { 6 x }$ 
B)  $\frac { 5 \sqrt { 6 x } } { x }$ 
C)  $\frac { 30 \sqrt { x } } { x }$ 
D)  $\frac { 30 \sqrt { 6 x } } { 6 x }$

Question 4

Use radical notation to write the expression. Simplify if possible.
-$( 6 x ) ^ { 2 / 5 }$

Accepted Answer

A)  $6 \sqrt [ 5 ] { \mathrm { x } ^ { 2 } }$ 
B)  $\sqrt [ 5 ] { 36 x ^ { 2 } }$ 
C)  $\sqrt [ 5 ] { 6 x ^ { 2 } }$ 
D)  $\sqrt [ 2 ] { 6 x ^ { 5 } }$ 
A)  $6 \sqrt [ 5 ] { \mathrm { x } ^ { 2 } }$ 
B)  $\sqrt [ 5 ] { 36 x ^ { 2 } }$ 
C)  $\sqrt [ 5 ] { 6 x ^ { 2 } }$ 
D)  $\sqrt [ 2 ] { 6 x ^ { 5 } }$

Question 5

Perform the indicated operation. Write the result in the form a + bi.
-$( 9 - 3 i ) - ( 9 - i )$

Accepted Answer

A)  $- 2 \mathrm { i }$ 
B)  $18 - 4 \mathrm { i }$ 
C)  $- 4 \mathrm { i }$ 
D)  $18 - 2 \mathrm { i }$ 
A)  $- 2 \mathrm { i }$ 
B)  $18 - 4 \mathrm { i }$ 
C)  $- 4 \mathrm { i }$ 
D)  $18 - 2 \mathrm { i }$

Question 6

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$$\sqrt [ 3 ] { 27 \mathrm {~m} ^ { 3 } } \cdot \sqrt [ 3 ] { 125 \mathrm {~m} ^ { 3 } }$$

Accepted Answer

A)  $- 3 \mathrm {~m}$ 
B)  $15 m ^ { 2 }$ 
C)  $8 \mathrm {~m}$ 
D)  15 
A)  $- 3 \mathrm {~m}$ 
B)  $15 m ^ { 2 }$ 
C)  $8 \mathrm {~m}$ 
D)  15

Question 7

Simplify. Assume that all variables represent positive real numbers.
-$$\sqrt [ 5 ] { 243 x ^ { 3 } y ^ { 29 } }$$

Accepted Answer

A)  $3 x y ^ { 4 } \sqrt [ 5 ] { x ^ { 2 } y ^ { 4 } }$ 
B)  $3 y ^ { 5 } \sqrt [ 5 ] { x ^ { 3 } y ^ { 4 } }$ 
C)  $3 x y ^ { 5 } \sqrt [ 5 ] { 3 x y ^ { 4 } }$ 
D)  $3 y ^ { 4 } \sqrt [ 5 ] { 3 x ^ { 3 } y ^ { 4 } }$ 
A)  $3 x y ^ { 4 } \sqrt [ 5 ] { x ^ { 2 } y ^ { 4 } }$ 
B)  $3 y ^ { 5 } \sqrt [ 5 ] { x ^ { 3 } y ^ { 4 } }$ 
C)  $3 x y ^ { 5 } \sqrt [ 5 ] { 3 x y ^ { 4 } }$ 
D)  $3 y ^ { 4 } \sqrt [ 5 ] { 3 x ^ { 3 } y ^ { 4 } }$

Question 8

Provide an appropriate response.
-Rationalize the numerator of $\frac { \sqrt { 11 } + x } { 5 }$ and simplify.

Accepted Answer

A)  $5 ( \sqrt { 11 } + x )$ 
B)  $\frac { 11 - x ^ { 2 } } { 5 ( \sqrt { 11 - x ) } }$ 
C)  $\frac { 11 - x ^ { 2 } } { \sqrt { 55 - 5 x } }$ 
D)  $\frac { 11 + x ^ { 2 } } { 5 ( \sqrt { 11 } + x ) }$ 
A)  $5 ( \sqrt { 11 } + x )$ 
B)  $\frac { 11 - x ^ { 2 } } { 5 ( \sqrt { 11 - x ) } }$ 
C)  $\frac { 11 - x ^ { 2 } } { \sqrt { 55 - 5 x } }$ 
D)  $\frac { 11 + x ^ { 2 } } { 5 ( \sqrt { 11 } + x ) }$

Question 9

Find the distance between the pair of points.
-$( 3 , - 2 )$ and $( 5 , - 6 )$

Accepted Answer

A)  $12 \sqrt { 3 }$ units 
B)  12 units 
C)  6 units 
D)  $2 \sqrt { 5 }$ units 
A)  $12 \sqrt { 3 }$ units 
B)  12 units 
C)  6 units 
D)  $2 \sqrt { 5 }$ units

Question 10

Simplify. Assume that all variables represent positive real numbers.
-$$\sqrt [ 3 ] { - 64 a ^ { 14 } b ^ { 10 } }$$

Accepted Answer

A)  $ 4 \sqrt{\mathrm{a}^{10} \mathrm{~b}^{14}} $ 
B)  $ 4 a^{2} b \sqrt[3]{a^{4} b^{3}} $ 
C)  $ -4 a^{4} b^{3} \sqrt[3]{a^{2} b} $ 
D)  $ 4 a b \sqrt[3]{a^{4} b^{5}} $ 
A)  $ 4 \sqrt{\mathrm{a}^{10} \mathrm{~b}^{14}} $ 
B)  $ 4 a^{2} b \sqrt[3]{a^{4} b^{3}} $ 
C)  $ -4 a^{4} b^{3} \sqrt[3]{a^{2} b} $ 
D)  $ 4 a b \sqrt[3]{a^{4} b^{5}} $

Question 11

Perform the indicated operation. Write the result in the form a + bi.
-$$( 2 + 8 i ) ^ { 2 }$$

Accepted Answer

A)  $4 + 32 i + 64 i ^ { 2 }$ 
B)  $68 + 32 i$ 
C)  $- 60 + 32 i$ 
D)  $- 60$ 
A)  $4 + 32 i + 64 i ^ { 2 }$ 
B)  $68 + 32 i$ 
C)  $- 60 + 32 i$ 
D)  $- 60$

Question 12

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 5 } { \mathrm { i } }$

Accepted Answer

A)  - 5 
B)  5 
C)  $5 \mathrm { i }$ 
D)  $- 5 \mathrm { i }$ 
A)  - 5 
B)  5 
C)  $5 \mathrm { i }$ 
D)  $- 5 \mathrm { i }$

Question 13

Find the root. Assume that all variables represent nonnegative real numbers.
-$\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }$

Accepted Answer

A)  $2 x ^ { 2 } y ^ { 4 }$ 
B)  $4 x ^ { 4 } y ^ { 2 }$ 
C)  $2.497 x ^ { 2 } y ^ { 4 }$ 
D)  not a real number 
A)  $2 x ^ { 2 } y ^ { 4 }$ 
B)  $4 x ^ { 4 } y ^ { 2 }$ 
C)  $2.497 x ^ { 2 } y ^ { 4 }$ 
D)  not a real number

Question 14

Find the root. Use absolute value bars when necessary.
-$$\sqrt [ 3 ] { ( - 3 ) ^ { 3 } }$$

Accepted Answer

A)  not a real number 
B)  $- 3$ 
C)  3 
D)  $- 27$ 
A)  not a real number 
B)  $- 3$ 
C)  3 
D)  $- 27$

Question 15

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt { y ^ { 9 } }$

Accepted Answer

A)  $y ^ { 8 } \sqrt { y }$ 
B)  $y ^ { 4 } \sqrt { y }$ 
C)  $\sqrt { \mathrm { y } ^ { 9 } }$ 
D)  $y \sqrt { y ^ { 7 } }$ 
A)  $y ^ { 8 } \sqrt { y }$ 
B)  $y ^ { 4 } \sqrt { y }$ 
C)  $\sqrt { \mathrm { y } ^ { 9 } }$ 
D)  $y \sqrt { y ^ { 7 } }$

Question 16

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$$\sqrt [ 4 ] { 81 } \cdot \sqrt [ 4 ] { 256 }$$

Accepted Answer

A)  $- 1$ 
B)  7 
C)  $- 12$ 
D)  12 
A)  $- 1$ 
B)  7 
C)  $- 12$ 
D)  12

Question 17

Simplify. Assume that all variables represent positive real numbers.
-$$\frac { \sqrt { 189 x ^ { 5 } y ^ { 6 } } } { \sqrt { 3 y ^ { 4 } } }$$

Accepted Answer

A)  $3 x ^ { 4 } y ^ { 2 } \sqrt { 7 x y }$ 
B)  $9 x ^ { 2 } y \sqrt { 7 x }$ 
C)  $3 x ^ { 2 } y \sqrt { 7 x }$ 
D)  $63 x y \sqrt { x }$ 
A)  $3 x ^ { 4 } y ^ { 2 } \sqrt { 7 x y }$ 
B)  $9 x ^ { 2 } y \sqrt { 7 x }$ 
C)  $3 x ^ { 2 } y \sqrt { 7 x }$ 
D)  $63 x y \sqrt { x }$

Question 18

Use the Pythagorean theorem to find the unknown side of the right triangle.
-

Accepted Answer

A)  $2 \sqrt { 7 }$ 
B)  $\sqrt { 22 }$ 
C)  22 
D)  4 
A)  $2 \sqrt { 7 }$ 
B)  $\sqrt { 22 }$ 
C)  22 
D)  4

Question 19

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( \sqrt { 2 x - 1 } + 3 ) ^ { 2 }$

Accepted Answer

A)  $2 x + 9 \sqrt { 2 x - 1 } + 9$ 
B)  $2 x + 6 \sqrt { 2 x - 1 } + 9$ 
C)  $2 x + 6 \sqrt { 2 x - 1 } + 8$ 
D)  $2 x + 9 \sqrt { 2 x - 1 } + 8$ 
A)  $2 x + 9 \sqrt { 2 x - 1 } + 9$ 
B)  $2 x + 6 \sqrt { 2 x - 1 } + 9$ 
C)  $2 x + 6 \sqrt { 2 x - 1 } + 8$ 
D)  $2 x + 9 \sqrt { 2 x - 1 } + 8$

Question 20

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive
exponents.
-$\left( \frac { 9 z ^ { 8 / 3 } } { x ^ { - 2 / 3 } y ^ { 7 / 3 } } \right) ^ { 1 / 2 }$

Accepted Answer

A)  $\frac { 9 z ^ { 4 / 3 } } { x ^ { 1 / 3 } y ^ { 7 / 6 } }$ 
B)  $\frac { 3 z ^ { 4 / 3 } x ^ { 1 / 3 } } { y ^ { 7 / 6 } }$ 
C)  $\frac { 3 x ^ { 2 / 3 } } { y ^ { 7 / 6 } }$ 
D)  $\frac { 3 z ^ { 4 / 7 } x ^ { 1 / 6 } } { y ^ { 7 / 6 } }$ 
A)  $\frac { 9 z ^ { 4 / 3 } } { x ^ { 1 / 3 } y ^ { 7 / 6 } }$ 
B)  $\frac { 3 z ^ { 4 / 3 } x ^ { 1 / 3 } } { y ^ { 7 / 6 } }$ 
C)  $\frac { 3 x ^ { 2 / 3 } } { y ^ { 7 / 6 } }$ 
D)  $\frac { 3 z ^ { 4 / 7 } x ^ { 1 / 6 } } { y ^ { 7 / 6 } }$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 5 a } - 2 \sqrt { 80 a } - 6 \sqrt { 125 a }$

Use the quotient rule to divide and simplify. - $\sqrt { \frac { 486 \mathrm { r } ^ { 2 } \mathrm { y } } { \mathrm { x } ^ { 4 } } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 30 } { \sqrt { 6 x } }$

Use radical notation to write the expression. Simplify if possible. - $( 6 x ) ^ { 2 / 5 }$

Perform the indicated operation. Write the result in the form a + bi. - $( 9 - 3 i ) - ( 9 - i )$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt [ 3 ] { 27 \mathrm {~m} ^ { 3 } } \cdot \sqrt [ 3 ] { 125 \mathrm {~m} ^ { 3 } }$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 5 ] { 243 x ^ { 3 } y ^ { 29 } }$

Provide an appropriate response. -Rationalize the numerator of $\frac { \sqrt { 11 } + x } { 5 }$ and simplify.

Find the distance between the pair of points. - $( 3 , - 2 )$ and $( 5 , - 6 )$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 3 ] { - 64 a ^ { 14 } b ^ { 10 } }$

Perform the indicated operation. Write the result in the form a + bi. - $( 2 + 8 i ) ^ { 2 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 5 } { \mathrm { i } }$

Find the root. Assume that all variables represent nonnegative real numbers. - $\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }$

Find the root. Use absolute value bars when necessary. - $\sqrt [ 3 ] { ( - 3 ) ^ { 3 } }$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt { y ^ { 9 } }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt [ 4 ] { 81 } \cdot \sqrt [ 4 ] { 256 }$

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 189 x ^ { 5 } y ^ { 6 } } } { \sqrt { 3 y ^ { 4 } } }$

Use the Pythagorean theorem to find the unknown side of the right triangle. -

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 2 x - 1 } + 3 ) ^ { 2 }$

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive exponents. - $\left( \frac { 9 z ^ { 8 / 3 } } { x ^ { - 2 / 3 } y ^ { 7 / 3 } } \right) ^ { 1 / 2 }$

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Add or subtract. Assume all variables represent positive real numbers. - 5a−280a−6125a\sqrt { 5 a } - 2 \sqrt { 80 a } - 6 \sqrt { 125 a }5a​−280a​−6125a​

Use the quotient rule to divide and simplify. - 486r2yx4\sqrt { \frac { 486 \mathrm { r } ^ { 2 } \mathrm { y } } { \mathrm { x } ^ { 4 } } }x4486r2y​​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 306x\frac { 30 } { \sqrt { 6 x } }6x​30​

Use radical notation to write the expression. Simplify if possible. - (6x)2/5( 6 x ) ^ { 2 / 5 }(6x)2/5

Perform the indicated operation. Write the result in the form a + bi. - (9−3i)−(9−i)( 9 - 3 i ) - ( 9 - i )(9−3i)−(9−i)

Use the product rule to multiply. Assume all variables represent positive real numbers. - 27 m33⋅125 m33\sqrt [ 3 ] { 27 \mathrm {~m} ^ { 3 } } \cdot \sqrt [ 3 ] { 125 \mathrm {~m} ^ { 3 } }327 m3​⋅3125 m3​

Simplify. Assume that all variables represent positive real numbers. - 243x3y295\sqrt [ 5 ] { 243 x ^ { 3 } y ^ { 29 } }5243x3y29​

Provide an appropriate response. -Rationalize the numerator of 11+x5\frac { \sqrt { 11 } + x } { 5 }511​+x​ and simplify.

Find the distance between the pair of points. - (3,−2)( 3 , - 2 )(3,−2) and (5,−6)( 5 , - 6 )(5,−6)

Simplify. Assume that all variables represent positive real numbers. - −64a14b103\sqrt [ 3 ] { - 64 a ^ { 14 } b ^ { 10 } }3−64a14b10​

Perform the indicated operation. Write the result in the form a + bi. - (2+8i)2( 2 + 8 i ) ^ { 2 }(2+8i)2

Perform the indicated operation. Write the result in the form a + bi. - 5i\frac { 5 } { \mathrm { i } }i5​

Find the root. Assume that all variables represent nonnegative real numbers. - 16x8y164\sqrt [ 4 ] { 16 x ^ { 8 } y ^ { 16 } }416x8y16​

Find the root. Use absolute value bars when necessary. - (−3)33\sqrt [ 3 ] { ( - 3 ) ^ { 3 } }3(−3)3​

Simplify. Assume that all variables represent positive real numbers. - y9\sqrt { y ^ { 9 } }y9​

Use the product rule to multiply. Assume all variables represent positive real numbers. - 814⋅2564\sqrt [ 4 ] { 81 } \cdot \sqrt [ 4 ] { 256 }481​⋅4256​

Simplify. Assume that all variables represent positive real numbers. - 189x5y63y4\frac { \sqrt { 189 x ^ { 5 } y ^ { 6 } } } { \sqrt { 3 y ^ { 4 } } }3y4​189x5y6​​