Question 1

Use radical notation to write the expression. Simplify if possible.
-$$\left( - 125 x ^ { 15 } \right) ^ { 1 / 3 }$$

Accepted Answer

A)  $- 5 \sqrt [ 5 ] { x }$ 
B)  $- 5 x ^ { 5 }$ 
C)  $- 5 x ^ { 15 }$ 
D)  $- 125 x ^ { 5 }$ 
A)  $- 5 \sqrt [ 5 ] { x }$ 
B)  $- 5 x ^ { 5 }$ 
C)  $- 5 x ^ { 15 }$ 
D)  $- 125 x ^ { 5 }$

Question 2

Solve.
-$$\sqrt { 2 x + 8 } = 9$$

Accepted Answer

A)  146 
B)  $\frac { 73 } { 2 }$ 
C)  $\frac { 2 } { 73 }$ 
D)  $\varnothing$ 
A)  146 
B)  $\frac { 73 } { 2 }$ 
C)  $\frac { 2 } { 73 }$ 
D)  $\varnothing$

Question 3

Solve.
-Find the area of the rectangle.

Accepted Answer

A)  $42 \sqrt { 7 } \mathrm { sq } . \mathrm { ft }$ 
B)  $294 \mathrm { sq } . \mathrm { ft }$ 
C)  $6 \sqrt { 7 } \mathrm { sq } . \mathrm { ft }$ 
D)  $42 \mathrm { sq } . \mathrm { ft }$ 
A)  $42 \sqrt { 7 } \mathrm { sq } . \mathrm { ft }$ 
B)  $294 \mathrm { sq } . \mathrm { ft }$ 
C)  $6 \sqrt { 7 } \mathrm { sq } . \mathrm { ft }$ 
D)  $42 \mathrm { sq } . \mathrm { ft }$

Question 4

Evaluate.
-If $f ( x ) = \sqrt [ 3 ] { x + 20 }$, find the value of $f ( 7 )$

Accepted Answer

A)  $3 \sqrt [ 3 ] { 3 }$ 
B)  $- 3$ 
C)  3 
D)  $3 \sqrt { 3 }$ 
A)  $3 \sqrt [ 3 ] { 3 }$ 
B)  $- 3$ 
C)  3 
D)  $3 \sqrt { 3 }$

Question 5

Simplify. Assume that all variables represent positive real numbers.
-$$\frac { \sqrt [ 3 ] { 24 a ^ { 4 } b ^ { 9 } } } { \sqrt [ 3 ] { 3 a } }$$

Accepted Answer

A)  $2 a ^ { 2 } b ^ { 6 }$ 
B)  $2 a b ^ { 6 } \sqrt [ 3 ] { 2 a ^ { 2 } }$ 
C)  $2 a b ^ { 3 }$ 
D)  $2 a b ^ { 3 } \sqrt [ 3 ] { 2 a b }$ 
A)  $2 a ^ { 2 } b ^ { 6 }$ 
B)  $2 a b ^ { 6 } \sqrt [ 3 ] { 2 a ^ { 2 } }$ 
C)  $2 a b ^ { 3 }$ 
D)  $2 a b ^ { 3 } \sqrt [ 3 ] { 2 a b }$

Question 6

Use the properties of exponents to simplify. Write with positive exponents.
-$\frac { y ^ { 1 / 8 } } { y ^ { 5 / 8 } }$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $- \mathrm { y } ^ { 1 / 2 }$ 
C)  $\frac { 1 } { y ^ { 1 / 2 } }$ 
D)  $y ^ { 5 / 16 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $- \mathrm { y } ^ { 1 / 2 }$ 
C)  $\frac { 1 } { y ^ { 1 / 2 } }$ 
D)  $y ^ { 5 / 16 }$

Question 7

Solve.
-$$\sqrt [ 3 ] { 4 x - 2 } = \sqrt [ 3 ] { x + 11 }$$

Accepted Answer

A)  $\frac { 9 } { 5 }$ 
B)  3 
C)  $\frac { 13 } { 3 }$ 
D)  $\frac { 13 } { 4 }$ 
A)  $\frac { 9 } { 5 }$ 
B)  3 
C)  $\frac { 13 } { 3 }$ 
D)  $\frac { 13 } { 4 }$

Question 8

Find the square root. Assume that all variables represent nonnegative real numbers.
-$\sqrt { \frac { 441 } { 64 } }$

Accepted Answer

A)  $\frac { 55 } { 8 }$ 
B)  $\frac { 7 } { 3 }$ 
C)  $\frac { 11 } { 4 }$ 
D)  $\frac { 21 } { 8 }$ 
A)  $\frac { 55 } { 8 }$ 
B)  $\frac { 7 } { 3 }$ 
C)  $\frac { 11 } { 4 }$ 
D)  $\frac { 21 } { 8 }$

Question 9

Use radical notation to write the expression. Simplify if possible.
-$32 ^ { 4 / 5 }$

Accepted Answer

A)  256 
B)  128 
C)  512 
D)  16 
A)  256 
B)  128 
C)  512 
D)  16

Question 10

Rationalize the numerator and simplify. Assume all variables represent positive real numbers.
-$\frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 3 } }$

Accepted Answer

A)  $\frac { 2 } { \sqrt [ 3 ] { 6 } }$ 
B)  $\frac { 2 } { \sqrt [ 3 ] { 12 } }$ 
C)  $\frac { \sqrt [ 3 ] { 18 } } { 3 }$ 
D)  $\frac { \sqrt [ 3 ] { 6 } } { 3 }$ 
A)  $\frac { 2 } { \sqrt [ 3 ] { 6 } }$ 
B)  $\frac { 2 } { \sqrt [ 3 ] { 12 } }$ 
C)  $\frac { \sqrt [ 3 ] { 18 } } { 3 }$ 
D)  $\frac { \sqrt [ 3 ] { 6 } } { 3 }$

Question 11

Use the properties of exponents to simplify. Write with positive exponents.
-$$\left( r ^ { 1 / 3 } \cdot s ^ { 1 / 3 } \right) ^ { 2 }$$

Accepted Answer

A)  $r ^ { 2 / 3 } s ^ { 2 / 3 }$ 
B)  $r ^ { 1 / 6 } { } _ { \mathrm { s } } ^ { 1 / 6 }$ 
C)  $r ^ { 1 / 9 } s ^ { 1 / 9 }$ 
D)  $\mathrm { r } ^ { 2 } \mathrm {~s} ^ { 2 }$ 
A)  $r ^ { 2 / 3 } s ^ { 2 / 3 }$ 
B)  $r ^ { 1 / 6 } { } _ { \mathrm { s } } ^ { 1 / 6 }$ 
C)  $r ^ { 1 / 9 } s ^ { 1 / 9 }$ 
D)  $\mathrm { r } ^ { 2 } \mathrm {~s} ^ { 2 }$

Question 12

Use rational exponents to simplify. Assume that all variables represent positive real numbers.
-$\sqrt [ 8 ] { x ^ { 6 } }$

Accepted Answer

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

Question 13

Simplify. Assume that all variables represent positive real numbers.
-$$\frac { \sqrt [ 3 ] { 625 x ^ { 17 } y ^ { 16 } } } { \sqrt [ 3 ] { 5 x ^ { 5 } y ^ { 2 } } }$$

Accepted Answer

A)  $25 x ^ { 6 } y ^ { 3 } \sqrt [ 3 ] { y }$ 
B)  $10 x ^ { 4 } y ^ { 5 }$ 
C)  $5 x ^ { 4 } y ^ { 4 } \sqrt [ 3 ] { y ^ { 2 } }$ 
D)  $5 x ^ { 3 } y ^ { 4 } \sqrt [ 3 ] { x ^ { 2 } }$ 
A)  $25 x ^ { 6 } y ^ { 3 } \sqrt [ 3 ] { y }$ 
B)  $10 x ^ { 4 } y ^ { 5 }$ 
C)  $5 x ^ { 4 } y ^ { 4 } \sqrt [ 3 ] { y ^ { 2 } }$ 
D)  $5 x ^ { 3 } y ^ { 4 } \sqrt [ 3 ] { x ^ { 2 } }$

Question 14

Simplify. Assume that all variables represent any real number.
-$\sqrt [ 3 ] { ( - 11 ) ^ { 3 } }$

Accepted Answer

A)  $- 33$ 
B)  11 
C)  $- 11$ 
D)  33 
A)  $- 33$ 
B)  11 
C)  $- 11$ 
D)  33

Question 15

Perform the indicated operation. Write the result in the form a + bi.
-$$( 6 + 3 i ) - ( - 4 + i )$$

Accepted Answer

A)  $- 10 - 2 \mathrm { i }$ 
B)  $10 + 2 \mathrm { i }$ 
C)  $10 - 2 \mathrm { i }$ 
D)  $2 + 4 i$ 
A)  $- 10 - 2 \mathrm { i }$ 
B)  $10 + 2 \mathrm { i }$ 
C)  $10 - 2 \mathrm { i }$ 
D)  $2 + 4 i$

Question 16

Solve.
-$$\sqrt { 9 x - 8 } - 8 = 0$$

Accepted Answer

A)  64 
B)  $\frac { 56 } { 9 }$ 
C)  8 
D)  $\varnothing$ 
A)  64 
B)  $\frac { 56 } { 9 }$ 
C)  8 
D)  $\varnothing$

Question 17

Find the root. Assume that all variables represent nonnegative real numbers.
-$\sqrt [ 11 ] { x ^ { 66 } }$

Accepted Answer

A)  $\frac { x ^ { 6 } } { 11 }$ 
B)  $6 x ^ { 6 }$ 
C)  $x ^ { 6 }$ 
D)  $\frac { 11 } { x ^ { 6 } }$ 
A)  $\frac { x ^ { 6 } } { 11 }$ 
B)  $6 x ^ { 6 }$ 
C)  $x ^ { 6 }$ 
D)  $\frac { 11 } { x ^ { 6 } }$

Question 18

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

Accepted Answer

A)  19 
B)  $\sqrt { 181 }$ 
C)  $\sqrt { 19 }$ 
D)  181 
A)  19 
B)  $\sqrt { 181 }$ 
C)  $\sqrt { 19 }$ 
D)  181

Question 19

Provide an appropriate response.
-Find the midpoint of the line segment whose endpoints are $\left( \frac { 1 } { 5 } , - \frac { 1 } { 3 } \right)$ and $\left( \frac { 2 } { 5 } , \frac { 1 } { 3 } \right)$.

Accepted Answer

A)  $\left( - \frac { 3 } { 5 } , 0 \right)$ 
B)  $\left( \frac { 3 } { 5 } , 0 \right)$ 
C)  $\left( \frac { 3 } { 10 } , 0 \right)$ 
D)  $\left( \frac { 3 } { 5 } , - \frac { 1 } { 6 } \right)$ 
A)  $\left( - \frac { 3 } { 5 } , 0 \right)$ 
B)  $\left( \frac { 3 } { 5 } , 0 \right)$ 
C)  $\left( \frac { 3 } { 10 } , 0 \right)$ 
D)  $\left( \frac { 3 } { 5 } , - \frac { 1 } { 6 } \right)$

Question 20

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 14 x } \cdot \sqrt { 11 y }$

Accepted Answer

A)  $\sqrt { 14 \mathrm { x } + 11 \mathrm { y } }$ 
B)  $\sqrt { 154 x y }$ 
C)  $\sqrt { 25 x y }$ 
D)  $154 x y$ 
A)  $\sqrt { 14 \mathrm { x } + 11 \mathrm { y } }$ 
B)  $\sqrt { 154 x y }$ 
C)  $\sqrt { 25 x y }$ 
D)  $154 x y$

Use radical notation to write the expression. Simplify if possible. - $\left( - 125 x ^ { 15 } \right) ^ { 1 / 3 }$

Solve. - $\sqrt { 2 x + 8 } = 9$

Solve. -Find the area of the rectangle.

Evaluate. -If $f ( x ) = \sqrt [ 3 ] { x + 20 }$ , find the value of $f ( 7 )$

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 24 a ^ { 4 } b ^ { 9 } } } { \sqrt [ 3 ] { 3 a } }$

Use the properties of exponents to simplify. Write with positive exponents. - $\frac { y ^ { 1 / 8 } } { y ^ { 5 / 8 } }$

Solve. - $\sqrt [ 3 ] { 4 x - 2 } = \sqrt [ 3 ] { x + 11 }$

Find the square root. Assume that all variables represent nonnegative real numbers. - $\sqrt { \frac { 441 } { 64 } }$

Use radical notation to write the expression. Simplify if possible. - $32 ^ { 4 / 5 }$

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 3 } }$

Use the properties of exponents to simplify. Write with positive exponents. - $\left( r ^ { 1 / 3 } \cdot s ^ { 1 / 3 } \right) ^ { 2 }$

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - $\sqrt [ 8 ] { x ^ { 6 } }$

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt [ 3 ] { 625 x ^ { 17 } y ^ { 16 } } } { \sqrt [ 3 ] { 5 x ^ { 5 } y ^ { 2 } } }$

Simplify. Assume that all variables represent any real number. - $\sqrt [ 3 ] { ( - 11 ) ^ { 3 } }$

Perform the indicated operation. Write the result in the form a + bi. - $( 6 + 3 i ) - ( - 4 + i )$

Solve. - $\sqrt { 9 x - 8 } - 8 = 0$

Find the root. Assume that all variables represent nonnegative real numbers. - $\sqrt [ 11 ] { x ^ { 66 } }$

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

Provide an appropriate response. -Find the midpoint of the line segment whose endpoints are $\left( \frac { 1 } { 5 } , - \frac { 1 } { 3 } \right)$ and $\left( \frac { 2 } { 5 } , \frac { 1 } { 3 } \right)$ .

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 14 x } \cdot \sqrt { 11 y }$

Real Numbers and Algebraic Expressions

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Exam 7: Rational Exponents, Radicals, and Complex Numbers

Use radical notation to write the expression. Simplify if possible. - (−125x15)1/3\left( - 125 x ^ { 15 } \right) ^ { 1 / 3 }(−125x15)1/3

Solve. - 2x+8=9\sqrt { 2 x + 8 } = 92x+8​=9

Solve. -Find the area of the rectangle.

Evaluate. -If f(x)=x+203f ( x ) = \sqrt [ 3 ] { x + 20 }f(x)=3x+20​ , find the value of f(7)f ( 7 )f(7)

Simplify. Assume that all variables represent positive real numbers. - 24a4b933a3\frac { \sqrt [ 3 ] { 24 a ^ { 4 } b ^ { 9 } } } { \sqrt [ 3 ] { 3 a } }33a​324a4b9​​

Use the properties of exponents to simplify. Write with positive exponents. - y1/8y5/8\frac { y ^ { 1 / 8 } } { y ^ { 5 / 8 } }y5/8y1/8​

Solve. - 4x−23=x+113\sqrt [ 3 ] { 4 x - 2 } = \sqrt [ 3 ] { x + 11 }34x−2​=3x+11​

Find the square root. Assume that all variables represent nonnegative real numbers. - 44164\sqrt { \frac { 441 } { 64 } }64441​​

Use radical notation to write the expression. Simplify if possible. - 324/532 ^ { 4 / 5 }324/5

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - 2333\frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 3 } }33​32​​

Use the properties of exponents to simplify. Write with positive exponents. - (r1/3⋅s1/3)2\left( r ^ { 1 / 3 } \cdot s ^ { 1 / 3 } \right) ^ { 2 }(r1/3⋅s1/3)2

Use rational exponents to simplify. Assume that all variables represent positive real numbers. - x68\sqrt [ 8 ] { x ^ { 6 } }8x6​

Simplify. Assume that all variables represent positive real numbers. - 625x17y1635x5y23\frac { \sqrt [ 3 ] { 625 x ^ { 17 } y ^ { 16 } } } { \sqrt [ 3 ] { 5 x ^ { 5 } y ^ { 2 } } }35x5y2​3625x17y16​​

Simplify. Assume that all variables represent any real number. - (−11)33\sqrt [ 3 ] { ( - 11 ) ^ { 3 } }3(−11)3​

Perform the indicated operation. Write the result in the form a + bi. - (6+3i)−(−4+i)( 6 + 3 i ) - ( - 4 + i )(6+3i)−(−4+i)

Solve. - 9x−8−8=0\sqrt { 9 x - 8 } - 8 = 09x−8​−8=0

Find the root. Assume that all variables represent nonnegative real numbers. - x6611\sqrt [ 11 ] { x ^ { 66 } }11x66​