Question 1

Add or subtract. Assume all variables represent positive real numbers.
-$$\sqrt { 2 } + 7 \sqrt { 8 } + 7 \sqrt { 18 }$$

Accepted Answer

A)  $14 \sqrt { 2 }$ 
B)  $14 \sqrt { 28 }$ 
C)  $36 \sqrt { 2 }$ 
D)  $36 \sqrt { 28 }$ 
A)  $14 \sqrt { 2 }$ 
B)  $14 \sqrt { 28 }$ 
C)  $36 \sqrt { 2 }$ 
D)  $36 \sqrt { 28 }$

Question 2

Use the properties of exponents to simplify. Write with positive exponents.
-$$\frac { y ^ { 3 / 4 } } { y ^ { 1 / 4 } }$$

Accepted Answer

A)  $y$ 
B)  $y ^ { 1 / 2 }$ 
C)  $y ^ { 3 / 4 }$ 
D)  $\frac { 1 } { y }$ 
A)  $y$ 
B)  $y ^ { 1 / 2 }$ 
C)  $y ^ { 3 / 4 }$ 
D)  $\frac { 1 } { y }$

Question 3

Perform the indicated operation. Write the result in the form a + bi.
-$7 \mathrm { i } + ( - 7 - \mathrm { i } )$

Accepted Answer

A)  $- 7 + 6 i$ 
B)  $- 7 + 8 \mathrm { i }$ 
C)  $7 - 8 \mathrm { i }$ 
D)  $7 - 6 \mathrm { i }$ 
A)  $- 7 + 6 i$ 
B)  $- 7 + 8 \mathrm { i }$ 
C)  $7 - 8 \mathrm { i }$ 
D)  $7 - 6 \mathrm { i }$

Question 4

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 9 } { \sqrt { 26 } + 5 }$

Accepted Answer

A)  $9 \sqrt { 26 } + 45$ 
B)  $\frac { 9 \sqrt { 26 } + 45 } { 52 }$ 
C)  $9 \sqrt { 26 } - 5$ 
D)  $9 \sqrt { 26 } - 45$ 
A)  $9 \sqrt { 26 } + 45$ 
B)  $\frac { 9 \sqrt { 26 } + 45 } { 52 }$ 
C)  $9 \sqrt { 26 } - 5$ 
D)  $9 \sqrt { 26 } - 45$

Question 5

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 9 \sqrt { 7 } } { \sqrt { 5 } }$

Accepted Answer

A)  $\frac { 9 \sqrt { 35 } } { 5 }$ 
B)  $\frac { \sqrt { 2835 } } { 5 }$ 
C)  $\frac { \sqrt { 35 } } { 45 }$ 
D)  $\frac { 45 \sqrt { 5 } } { 7 }$ 
A)  $\frac { 9 \sqrt { 35 } } { 5 }$ 
B)  $\frac { \sqrt { 2835 } } { 5 }$ 
C)  $\frac { \sqrt { 35 } } { 45 }$ 
D)  $\frac { 45 \sqrt { 5 } } { 7 }$

Question 6

Use the quotient rule to divide and simplify.
-$$\sqrt { \frac { 125 } { 49 } }$$

Accepted Answer

A)  $\frac { 5 } { 7 }$ 
B)  $25 \sqrt { 5 }$ 
C)  $\sqrt { \frac { 125 } { 7 } }$ 
D)  $\frac { 5 \sqrt { 5 } } { 7 }$ 
A)  $\frac { 5 } { 7 }$ 
B)  $25 \sqrt { 5 }$ 
C)  $\sqrt { \frac { 125 } { 7 } }$ 
D)  $\frac { 5 \sqrt { 5 } } { 7 }$

Question 7

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 4 ] { \frac { 81 } { 25 x ^ { 15 } } }$

Accepted Answer

A)  $\frac { 3 \sqrt [ 4 ] { 25 x ^ { 3 } } } { 5 x ^ { 4 } }$ 
B)  $\frac { 3 \sqrt [ 4 ] { 25 x } } { 5 x ^ { 3 } }$ 
C)  $\frac { 3 \sqrt [ 4 ] { 25 x } } { 5 x ^ { 4 } }$ 
D)  $\frac { 3 \sqrt [ 4 ] { 25 x ^ { 3 } } } { 5 x ^ { 3 } }$ 
A)  $\frac { 3 \sqrt [ 4 ] { 25 x ^ { 3 } } } { 5 x ^ { 4 } }$ 
B)  $\frac { 3 \sqrt [ 4 ] { 25 x } } { 5 x ^ { 3 } }$ 
C)  $\frac { 3 \sqrt [ 4 ] { 25 x } } { 5 x ^ { 4 } }$ 
D)  $\frac { 3 \sqrt [ 4 ] { 25 x ^ { 3 } } } { 5 x ^ { 3 } }$

Question 8

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

Accepted Answer

A)  $8 - 4 \mathrm { i }$ 
B)  12 
C)  $8 - 2 i \sqrt { 2 }$ 
D)  4 
A)  $8 - 4 \mathrm { i }$ 
B)  12 
C)  $8 - 2 i \sqrt { 2 }$ 
D)  4

Question 9

Add or subtract. Assume all variables represent positive real numbers.
-$\sqrt { 48 } - \sqrt { 147 }$

Accepted Answer

A)  $- 3 \sqrt { 6 }$ 
B)  $11 \sqrt { 3 }$ 
C)  $- 9$ 
D)  $- 3 \sqrt { 3 }$ 
A)  $- 3 \sqrt { 6 }$ 
B)  $11 \sqrt { 3 }$ 
C)  $- 9$ 
D)  $- 3 \sqrt { 3 }$

Question 10

Solve.
-$$\sqrt { 5 x - 3 } = \sqrt { x + 6 }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $\frac { 9 } { 4 }$ 
D)  $\frac { 9 } { 5 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $\frac { 9 } { 4 }$ 
D)  $\frac { 9 } { 5 }$

Question 11

Find the cube root.
-$\sqrt [ 3 ] { \frac { 1 } { 64 } }$

Accepted Answer

A)  $\frac { 1 } { \sqrt [ 3 ] { 4 } }$ 
B)  $\frac { 1 } { 16 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  4 
A)  $\frac { 1 } { \sqrt [ 3 ] { 4 } }$ 
B)  $\frac { 1 } { 16 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  4

Question 12

Find the midpoint of the line segment whose endpoints are given.
-$( 4,6 ) , ( 1 , - 4 )$

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , 5 \right)$ 
B)  $\left( \frac { 5 } { 2 } , 1 \right)$ 
C)  $( 5,2 )$ 
D)  $( 3,10 )$ 
A)  $\left( \frac { 3 } { 2 } , 5 \right)$ 
B)  $\left( \frac { 5 } { 2 } , 1 \right)$ 
C)  $( 5,2 )$ 
D)  $( 3,10 )$

Question 13

Simplify. Assume that all variables represent positive real numbers.
-$\sqrt { 50 x ^ { 2 } y }$

Accepted Answer

A)  $5 x \sqrt { 2 y }$ 
B)  $5 x y ^ { 2 } \sqrt { 2 }$ 
C)  $5 x ^ { 2 } \sqrt { 2 y }$ 
D)  $5 x y \sqrt { 2 }$ 
A)  $5 x \sqrt { 2 y }$ 
B)  $5 x y ^ { 2 } \sqrt { 2 }$ 
C)  $5 x ^ { 2 } \sqrt { 2 y }$ 
D)  $5 x y \sqrt { 2 }$

Question 14

Add or subtract. Assume all variables represent positive real numbers.
-$$2 \sqrt [ 4 ] { 6 } + 3 \sqrt [ 3 ] { 6 }$$

Accepted Answer

A)  $5 \sqrt [ 3 ] { 6 }$ 
B)  $5 \sqrt [ 4 ] { 6 }$ 
C)  $6 \sqrt [ 4 ] { 36 }$ 
D)  $2 \sqrt [ 4 ] { 6 } + 3 \sqrt [ 3 ] { 6 }$ 
A)  $5 \sqrt [ 3 ] { 6 }$ 
B)  $5 \sqrt [ 4 ] { 6 }$ 
C)  $6 \sqrt [ 4 ] { 36 }$ 
D)  $2 \sqrt [ 4 ] { 6 } + 3 \sqrt [ 3 ] { 6 }$

Question 15

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
-$\frac { 5 \sqrt { 2 } + \sqrt { 14 } } { 5 \sqrt { 2 } - \sqrt { 14 } }$

Accepted Answer

A)  $\frac { 64 + 10 \sqrt { 28 } } { 36 }$ 
B)  $\frac { 32 + 10 \sqrt { 7 } } { 18 }$ 
C)  $\frac { 64 + 20 \sqrt { 7 } } { 36 }$ 
D)  $\frac { 16 + 5 \sqrt { 7 } } { 9 }$ 
A)  $\frac { 64 + 10 \sqrt { 28 } } { 36 }$ 
B)  $\frac { 32 + 10 \sqrt { 7 } } { 18 }$ 
C)  $\frac { 64 + 20 \sqrt { 7 } } { 36 }$ 
D)  $\frac { 16 + 5 \sqrt { 7 } } { 9 }$

Question 16

Use the quotient rule to divide and simplify.
-$\sqrt { \frac { 42 } { x ^ { 4 } } }$

Accepted Answer

A)  $\frac { \sqrt { 42 } } { x ^ { 2 } }$ 
B)  $\frac { \sqrt { 42 } } { \sqrt { x ^ { 4 } } }$ 
C)  $\frac { \sqrt { 42 x ^ { 4 } } } { x ^ { 4 } }$ 
D)  $\frac { \sqrt { 42 } } { x }$ 
A)  $\frac { \sqrt { 42 } } { x ^ { 2 } }$ 
B)  $\frac { \sqrt { 42 } } { \sqrt { x ^ { 4 } } }$ 
C)  $\frac { \sqrt { 42 x ^ { 4 } } } { x ^ { 4 } }$ 
D)  $\frac { \sqrt { 42 } } { x }$

Question 17

Use rational exponents to simplify. Assume that all variables represent positive real numbers.
-$$\sqrt [ 12 ] { 8 \mathrm { x } ^ { 3 } }$$

Accepted Answer

A)  $\sqrt [ 4 ] { 2 x }$ 
B)  $2 \sqrt [ 4 ] { 2 x }$ 
C)  $\sqrt [ 3 ] { 2 x }$ 
D)  $\frac { 1 } { 16 x }$ 
A)  $\sqrt [ 4 ] { 2 x }$ 
B)  $2 \sqrt [ 4 ] { 2 x }$ 
C)  $\sqrt [ 3 ] { 2 x }$ 
D)  $\frac { 1 } { 16 x }$

Question 18

Write with positive exponents. Simplify if possible.
-$$- 243 ^ { - 4 / 5 }$$

Accepted Answer

A)  $\frac { 1 } { 81 }$ 
B)  81 
C)  $- \frac { 1 } { 81 }$ 
D)  not a real number 
A)  $\frac { 1 } { 81 }$ 
B)  81 
C)  $- \frac { 1 } { 81 }$ 
D)  not a real number

Question 19

Use radical notation to write the expression. Simplify if possible.
-$( - 8 ) ^ { 4 / 3 }$

Accepted Answer

A)  $- 16$ 
B)  16 
C)  64 
D)  not a real number 
A)  $- 16$ 
B)  16 
C)  64 
D)  not a real number

Question 20

Perform the indicated operation. Write the result in the form a + bi.
-$\frac { 5 + 7 i } { 5 - 7 i }$

Accepted Answer

A)  $- \frac { 37 } { 12 } + \frac { 35 } { 12 } \mathrm { i }$ 
B)  $1 - \frac { 35 } { 12 } \mathrm { i }$ 
C)  $- \frac { 12 } { 37 } + \frac { 35 } { 37 } \mathrm { i }$ 
D)  $1 + \frac { 35 } { 37 } \mathrm { i }$ 
A)  $- \frac { 37 } { 12 } + \frac { 35 } { 12 } \mathrm { i }$ 
B)  $1 - \frac { 35 } { 12 } \mathrm { i }$ 
C)  $- \frac { 12 } { 37 } + \frac { 35 } { 37 } \mathrm { i }$ 
D)  $1 + \frac { 35 } { 37 } \mathrm { i }$

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 2 } + 7 \sqrt { 8 } + 7 \sqrt { 18 }$

Use the properties of exponents to simplify. Write with positive exponents. - $\frac { y ^ { 3 / 4 } } { y ^ { 1 / 4 } }$

Perform the indicated operation. Write the result in the form a + bi. - $7 \mathrm { i } + ( - 7 - \mathrm { i } )$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 9 } { \sqrt { 26 } + 5 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 9 \sqrt { 7 } } { \sqrt { 5 } }$

Use the quotient rule to divide and simplify. - $\sqrt { \frac { 125 } { 49 } }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 4 ] { \frac { 81 } { 25 x ^ { 15 } } }$

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

Add or subtract. Assume all variables represent positive real numbers. - $\sqrt { 48 } - \sqrt { 147 }$

Solve. - $\sqrt { 5 x - 3 } = \sqrt { x + 6 }$

Find the cube root. - $\sqrt [ 3 ] { \frac { 1 } { 64 } }$

Find the midpoint of the line segment whose endpoints are given. - $( 4,6 ) , ( 1 , - 4 )$

Simplify. Assume that all variables represent positive real numbers. - $\sqrt { 50 x ^ { 2 } y }$

Add or subtract. Assume all variables represent positive real numbers. - $2 \sqrt [ 4 ] { 6 } + 3 \sqrt [ 3 ] { 6 }$

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - $\frac { 5 \sqrt { 2 } + \sqrt { 14 } } { 5 \sqrt { 2 } - \sqrt { 14 } }$

Use the quotient rule to divide and simplify. - $\sqrt { \frac { 42 } { x ^ { 4 } } }$

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 12 ] { 8 \mathrm { x } ^ { 3 } }$

Write with positive exponents. Simplify if possible. - $- 243 ^ { - 4 / 5 }$

Use radical notation to write the expression. Simplify if possible. - $( - 8 ) ^ { 4 / 3 }$

Perform the indicated operation. Write the result in the form a + bi. - $\frac { 5 + 7 i } { 5 - 7 i }$

Real Numbers and Algebraic Expressions

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Add or subtract. Assume all variables represent positive real numbers. - 2+78+718\sqrt { 2 } + 7 \sqrt { 8 } + 7 \sqrt { 18 }2​+78​+718​

Use the properties of exponents to simplify. Write with positive exponents. - y3/4y1/4\frac { y ^ { 3 / 4 } } { y ^ { 1 / 4 } }y1/4y3/4​

Perform the indicated operation. Write the result in the form a + bi. - 7i+(−7−i)7 \mathrm { i } + ( - 7 - \mathrm { i } )7i+(−7−i)

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 926+5\frac { 9 } { \sqrt { 26 } + 5 }26​+59​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 975\frac { 9 \sqrt { 7 } } { \sqrt { 5 } }5​97​​

Use the quotient rule to divide and simplify. - 12549\sqrt { \frac { 125 } { 49 } }49125​​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 8125x154\sqrt [ 4 ] { \frac { 81 } { 25 x ^ { 15 } } }425x1581​​

Perform the indicated operation. Write the result in the form a + bi. - (8+2i)(8−2i)( \sqrt { 8 } + 2 \mathrm { i } ) ( \sqrt { 8 } - 2 \mathrm { i } )(8​+2i)(8​−2i)

Add or subtract. Assume all variables represent positive real numbers. - 48−147\sqrt { 48 } - \sqrt { 147 }48​−147​

Solve. - 5x−3=x+6\sqrt { 5 x - 3 } = \sqrt { x + 6 }5x−3​=x+6​

Find the cube root. - 1643\sqrt [ 3 ] { \frac { 1 } { 64 } }3641​​

Find the midpoint of the line segment whose endpoints are given. - (4,6),(1,−4)( 4,6 ) , ( 1 , - 4 )(4,6),(1,−4)

Simplify. Assume that all variables represent positive real numbers. - 50x2y\sqrt { 50 x ^ { 2 } y }50x2y​

Add or subtract. Assume all variables represent positive real numbers. - 264+3632 \sqrt [ 4 ] { 6 } + 3 \sqrt [ 3 ] { 6 }246​+336​

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. - 52+1452−14\frac { 5 \sqrt { 2 } + \sqrt { 14 } } { 5 \sqrt { 2 } - \sqrt { 14 } }52​−14​52​+14​​

Use the quotient rule to divide and simplify. - 42x4\sqrt { \frac { 42 } { x ^ { 4 } } }x442​​

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - 8x312\sqrt [ 12 ] { 8 \mathrm { x } ^ { 3 } }128x3​

Write with positive exponents. Simplify if possible. - −243−4/5- 243 ^ { - 4 / 5 }−243−4/5

Use radical notation to write the expression. Simplify if possible. - (−8)4/3( - 8 ) ^ { 4 / 3 }(−8)4/3

Perform the indicated operation. Write the result in the form a + bi. - 5+7i5−7i\frac { 5 + 7 i } { 5 - 7 i }5−7i5+7i​