Question 1

Use the product rule to multiply. Assume all variables represent positive real numbers.
-$\sqrt { 6 } \cdot \sqrt { 7 }$

Accepted Answer

A)  $\sqrt { 42 }$ 
B)  $\sqrt { 13 }$ 
C)  42 
D)  $\sqrt { 6 + 7 }$ 
A)  $\sqrt { 42 }$ 
B)  $\sqrt { 13 }$ 
C)  42 
D)  $\sqrt { 6 + 7 }$

Question 2

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )$

Accepted Answer

A)  $7 - 2 \sqrt { 5 }$ 
B)  2 
C)  32 
D)  $- 18$ 
A)  $7 - 2 \sqrt { 5 }$ 
B)  2 
C)  32 
D)  $- 18$

Question 3

Multiply, and then simplify if possible. Assume all variables represent positive real numbers.
-$$( 4 \sqrt { 3 } - 3 ) ^ { 2 }$$

Accepted Answer

A)  $39 - 24 \sqrt { 3 }$ 
B)  $57 - 24 \sqrt { 3 }$ 
C)  $45 - 24 \sqrt { 3 }$ 
D)  $57 + 24 \sqrt { 3 }$ 
A)  $39 - 24 \sqrt { 3 }$ 
B)  $57 - 24 \sqrt { 3 }$ 
C)  $45 - 24 \sqrt { 3 }$ 
D)  $57 + 24 \sqrt { 3 }$

Question 4

Write in terms of i.
-$\sqrt { - 81 }$

Accepted Answer

A)  $9 \mathrm { i }$ 
B)  $\pm 9$ 
C)  $- 9 \mathrm { i }$ 
D)  $\mathrm { i } \sqrt { 9 }$ 
A)  $9 \mathrm { i }$ 
B)  $\pm 9$ 
C)  $- 9 \mathrm { i }$ 
D)  $\mathrm { i } \sqrt { 9 }$

Question 5

Provide an appropriate response.
-Find x.

Accepted Answer

A)  $x = \frac { 5 \sqrt { 2 } } { 2 }$ in. 
B)  $x = 5 \sqrt { 2 } \mathrm { in }$. 
C)  $x = \frac { \sqrt { 10 } } { 2 }$ in. 
D)  $x = 5$ in. 
A)  $x = \frac { 5 \sqrt { 2 } } { 2 }$ in. 
B)  $x = 5 \sqrt { 2 } \mathrm { in }$. 
C)  $x = \frac { \sqrt { 10 } } { 2 }$ in. 
D)  $x = 5$ in.

Question 6

Perform the indicated operation. Write the result in the form a + bi.
-$( - 9 + 2 i ) - 4$

Accepted Answer

A)  $- 13 + 2 \mathrm { i }$ 
B)  $- 5 - 2 \mathrm { i }$ 
C)  $13 - 2 \mathrm { i }$ 
D)  $- 5 + 2 i$ 
A)  $- 13 + 2 \mathrm { i }$ 
B)  $- 5 - 2 \mathrm { i }$ 
C)  $13 - 2 \mathrm { i }$ 
D)  $- 5 + 2 i$

Question 7

Write in terms of i.
-$\sqrt { - 81 }$

Accepted Answer

A)  $- 9 \mathrm { i }$ 
B)  $- \mathrm { i } \sqrt { 9 }$ 
C)  $9 \mathrm { i }$ 
D)  $\pm 9$ 
A)  $- 9 \mathrm { i }$ 
B)  $- \mathrm { i } \sqrt { 9 }$ 
C)  $9 \mathrm { i }$ 
D)  $\pm 9$

Question 8

Add or subtract. Assume all variables represent positive real numbers.
-$$10 \sqrt [ 4 ] { x ^ { 7 } } - 3 x \sqrt [ 4 ] { x ^ { 3 } }$$

Accepted Answer

A)  $ 13 \sqrt[4]{x^{3}} $ 
B)  $ 7 x \sqrt[4]{x^{3}} $ 
C)  $ 10 \sqrt[4]{x^{7}}-3 x \sqrt[4]{x^{3}} $ 
D)  $ 7 \times \sqrt[4]{x^{7}} $ 
A)  $ 13 \sqrt[4]{x^{3}} $ 
B)  $ 7 x \sqrt[4]{x^{3}} $ 
C)  $ 10 \sqrt[4]{x^{7}}-3 x \sqrt[4]{x^{3}} $ 
D)  $ 7 \times \sqrt[4]{x^{7}} $

Question 9

Use radical notation to write the expression. Simplify if possible.
-$$- 243 ^ { 1 / 5 }$$

Accepted Answer

A)  81 
B)  243 
C)  -3 
D)  -27 
A)  81 
B)  243 
C)  -3 
D)  -27

Question 10

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive
exponents.
-$$\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }$$

Accepted Answer

A)  $3 \sqrt { a ^ { 13 } b ^ { 11 } }$ 
B)  $3 a ^ { 2 } b \sqrt [ 3 ] { a ^ { 3 } b ^ { 4 } }$ 
C)  $3 a b \sqrt [ 3 ] { a ^ { 5 } b ^ { 4 } }$ 
D)  $- 3 a ^ { 3 } b ^ { 4 } \sqrt [ 3 ] { a ^ { 2 } b }$ 
A)  $3 \sqrt { a ^ { 13 } b ^ { 11 } }$ 
B)  $3 a ^ { 2 } b \sqrt [ 3 ] { a ^ { 3 } b ^ { 4 } }$ 
C)  $3 a b \sqrt [ 3 ] { a ^ { 5 } b ^ { 4 } }$ 
D)  $- 3 a ^ { 3 } b ^ { 4 } \sqrt [ 3 ] { a ^ { 2 } b }$

Question 11

Perform the indicated operation. Assume that all variables represent positive real numbers.
-$( \sqrt { x } - 14 ) ^ { 2 }$

Accepted Answer

A)  $x + 196$ 
B)  $x + 28 \sqrt { x } + 196$ 
C)  $x - 28 \sqrt { x } + 196$ 
D)  $x - 196$ 
A)  $x + 196$ 
B)  $x + 28 \sqrt { x } + 196$ 
C)  $x - 28 \sqrt { x } + 196$ 
D)  $x - 196$

Question 12

Write with positive exponents. Simplify if possible.
-$$\frac { 1 } { x ^ { - 8 / 3 } }$$

Accepted Answer

A)  $\frac { - 1 } { x ^ { 8 / 3 } }$ 
B)  $\frac { 1 } { x ^ { 3 / 8 } }$ 
C)  $- x ^ { 8 / 3 }$ 
D)  $x ^ { 8 / 3 }$ 
A)  $\frac { - 1 } { x ^ { 8 / 3 } }$ 
B)  $\frac { 1 } { x ^ { 3 / 8 } }$ 
C)  $- x ^ { 8 / 3 }$ 
D)  $x ^ { 8 / 3 }$

Question 13

Solve.
-A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut while the balloon is directly above a flag pole that is 40 feet from where the rope is staked down. Find the altitude of the balloon if the rope is 50 feet long.

Accepted Answer

A)  $10 \sqrt { 41 } \mathrm { ft }$ 
B)  $3 \sqrt { 10 } \mathrm { ft }$ 
C)  $\sqrt { 10 } \mathrm { ft }$ 
D)  $30 \mathrm { ft }$ 
A)  $10 \sqrt { 41 } \mathrm { ft }$ 
B)  $3 \sqrt { 10 } \mathrm { ft }$ 
C)  $\sqrt { 10 } \mathrm { ft }$ 
D)  $30 \mathrm { ft }$

Question 14

Evaluate.
-If $f ( x ) = \sqrt { 2 x - 7 }$, find the value of $f ( 11 )$.

Accepted Answer

A)  $\sqrt { 15 }$ 
B)  22 
C)  15 
D)  $\sqrt { 22 }$ 
A)  $\sqrt { 15 }$ 
B)  22 
C)  15 
D)  $\sqrt { 22 }$

Question 15

Simplify. Assume that all variables represent positive real numbers.
-$\frac { \sqrt { 168 } } { \sqrt { 6 } }$

Accepted Answer

A)  $\frac { \sqrt { 1008 } } { 6 }$ 
B)  $2 \sqrt { 7 }$ 
C)  6 
D)  $\frac { \sqrt { 168 } } { 6 }$ 
A)  $\frac { \sqrt { 1008 } } { 6 }$ 
B)  $2 \sqrt { 7 }$ 
C)  6 
D)  $\frac { \sqrt { 168 } } { 6 }$

Question 16

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

Accepted Answer

A)  $\frac { 2 } { \sqrt { 2 } }$ 
B)  $\frac { 5 } { \sqrt { 10 } }$ 
C)  $\frac { 5 } { \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 10 } } { 2 }$ 
A)  $\frac { 2 } { \sqrt { 2 } }$ 
B)  $\frac { 5 } { \sqrt { 10 } }$ 
C)  $\frac { 5 } { \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 10 } } { 2 }$

Question 17

Simplify. Assume that all variables represent any real number.
-$\sqrt { 81 x ^ { 2 } }$

Accepted Answer

A)  $9 | \times |$ 
B)  $9 x$ 
C)  $- 9 | x |$ 
D)  $- 9 x$ 
A)  $9 | \times |$ 
B)  $9 x$ 
C)  $- 9 | x |$ 
D)  $- 9 x$

Question 18

Find the cube root.
-$$\sqrt [ 3 ] { 512 x ^ { 6 } }$$

Accepted Answer

A)  $22 x ^ { 2 }$ 
B)  $8 x ^ { 2 }$ 
C)  $64 x ^ { 2 }$ 
D)  not a real number 
A)  $22 x ^ { 2 }$ 
B)  $8 x ^ { 2 }$ 
C)  $64 x ^ { 2 }$ 
D)  not a real number

Question 19

Use the properties of exponents to simplify. Write with positive exponents.
-$\frac { 12 ^ { 1 / 7 } } { 12 ^ { - 13 / 7 } }$

Accepted Answer

A)  2 
B)  144 
C)  $12 ^ { 1 / 2 }$ 
D)  $\frac { 1 } { 12 ^ { 12 / 7 } }$ 
A)  2 
B)  144 
C)  $12 ^ { 1 / 2 }$ 
D)  $\frac { 1 } { 12 ^ { 12 / 7 } }$

Question 20

Add or subtract. Assume all variables represent positive real numbers.
-$$\frac { \sqrt [ 4 ] { 3750 } } { 11 x } - \frac { 2 \sqrt [ 4 ] { 6 } } { 22 x }$$

Accepted Answer

A)  $\frac { 4 \sqrt [ 4 ] { 6 } } { 22 x }$ 
B)  $\frac { 4 \sqrt [ 4 ] { 6 } } { 11 \mathrm { x } }$ 
C)  $\frac { 5 \sqrt [ 4 ] { 5 } } { x }$ 
D)  $- \frac { 6 \sqrt { 6 } } { 11 x }$ 
A)  $\frac { 4 \sqrt [ 4 ] { 6 } } { 22 x }$ 
B)  $\frac { 4 \sqrt [ 4 ] { 6 } } { 11 \mathrm { x } }$ 
C)  $\frac { 5 \sqrt [ 4 ] { 5 } } { x }$ 
D)  $- \frac { 6 \sqrt { 6 } } { 11 x }$

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 6 } \cdot \sqrt { 7 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( 4 \sqrt { 3 } - 3 ) ^ { 2 }$

Write in terms of i. - $\sqrt { - 81 }$

Provide an appropriate response. -Find x.

Perform the indicated operation. Write the result in the form a + bi. - $( - 9 + 2 i ) - 4$

Write in terms of i. - $\sqrt { - 81 }$

Add or subtract. Assume all variables represent positive real numbers. - $10 \sqrt [ 4 ] { x ^ { 7 } } - 3 x \sqrt [ 4 ] { x ^ { 3 } }$

Use radical notation to write the expression. Simplify if possible. - $- 243 ^ { 1 / 5 }$

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive exponents. - $\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }$

Perform the indicated operation. Assume that all variables represent positive real numbers. - $( \sqrt { x } - 14 ) ^ { 2 }$

Write with positive exponents. Simplify if possible. - $\frac { 1 } { x ^ { - 8 / 3 } }$

Solve. -A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut while the balloon is directly above a flag pole that is 40 feet from where the rope is staked down. Find the altitude of the balloon if the rope is 50 feet long.

Evaluate. -If $f ( x ) = \sqrt { 2 x - 7 }$ , find the value of $f ( 11 )$ .

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 168 } } { \sqrt { 6 } }$

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

Simplify. Assume that all variables represent any real number. - $\sqrt { 81 x ^ { 2 } }$

Find the cube root. - $\sqrt [ 3 ] { 512 x ^ { 6 } }$

Use the properties of exponents to simplify. Write with positive exponents. - $\frac { 12 ^ { 1 / 7 } } { 12 ^ { - 13 / 7 } }$

Add or subtract. Assume all variables represent positive real numbers. - $\frac { \sqrt [ 4 ] { 3750 } } { 11 x } - \frac { 2 \sqrt [ 4 ] { 6 } } { 22 x }$

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Rational Expressions

Quadratic Equations and Functions

Exponential and Logarithmic Functions

Conic Sections

Fractions

Filters

Exam 7: Rational Exponents, Radicals, and Complex Numbers

Use the product rule to multiply. Assume all variables represent positive real numbers. - 6⋅7\sqrt { 6 } \cdot \sqrt { 7 }6​⋅7​

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (7+5)(7−5)( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )(7​+5)(7​−5)

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - (43−3)2( 4 \sqrt { 3 } - 3 ) ^ { 2 }(43​−3)2

Write in terms of i. - −81\sqrt { - 81 }−81​

Provide an appropriate response. -Find x.

Perform the indicated operation. Write the result in the form a + bi. - (−9+2i)−4( - 9 + 2 i ) - 4(−9+2i)−4

Write in terms of i. - −81\sqrt { - 81 }−81​

Add or subtract. Assume all variables represent positive real numbers. - 10x74−3xx3410 \sqrt [ 4 ] { x ^ { 7 } } - 3 x \sqrt [ 4 ] { x ^ { 3 } }104x7​−3x4x3​

Use radical notation to write the expression. Simplify if possible. - −2431/5- 243 ^ { 1 / 5 }−2431/5

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive exponents. - −27a11b133\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }3−27a11b13​

Perform the indicated operation. Assume that all variables represent positive real numbers. - (x−14)2( \sqrt { x } - 14 ) ^ { 2 }(x​−14)2

Write with positive exponents. Simplify if possible. - 1x−8/3\frac { 1 } { x ^ { - 8 / 3 } }x−8/31​

Solve. -A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut while the balloon is directly above a flag pole that is 40 feet from where the rope is staked down. Find the altitude of the balloon if the rope is 50 feet long.

Evaluate. -If f(x)=2x−7f ( x ) = \sqrt { 2 x - 7 }f(x)=2x−7​ , find the value of f(11)f ( 11 )f(11) .

Simplify. Assume that all variables represent positive real numbers. - 1686\frac { \sqrt { 168 } } { \sqrt { 6 } }6​168​​

Rationalize the numerator and simplify. Assume all variables represent positive real numbers. - 52\sqrt { \frac { 5 } { 2 } }25​​

Simplify. Assume that all variables represent any real number. - 81x2\sqrt { 81 x ^ { 2 } }81x2​

Find the cube root. - 512x63\sqrt [ 3 ] { 512 x ^ { 6 } }3512x6​

Use the properties of exponents to simplify. Write with positive exponents. - 121/712−13/7\frac { 12 ^ { 1 / 7 } } { 12 ^ { - 13 / 7 } }12−13/7121/7​

Add or subtract. Assume all variables represent positive real numbers. - 3750411x−26422x\frac { \sqrt [ 4 ] { 3750 } } { 11 x } - \frac { 2 \sqrt [ 4 ] { 6 } } { 22 x }11x43750​​−22x246​​

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Rational Expressions

Quadratic Equations and Functions

Exponential and Logarithmic Functions

Conic Sections

Fractions

Filters

Use the product rule to multiply. Assume all variables represent positive real numbers. - $\sqrt { 6 } \cdot \sqrt { 7 }$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( \sqrt { 7 } + 5 ) ( \sqrt { 7 } - 5 )$

Multiply, and then simplify if possible. Assume all variables represent positive real numbers. - $( 4 \sqrt { 3 } - 3 ) ^ { 2 }$

Write in terms of i. - $\sqrt { - 81 }$

Perform the indicated operation. Write the result in the form a + bi. - $( - 9 + 2 i ) - 4$

Write in terms of i. - $\sqrt { - 81 }$

Add or subtract. Assume all variables represent positive real numbers. - $10 \sqrt [ 4 ] { x ^ { 7 } } - 3 x \sqrt [ 4 ] { x ^ { 3 } }$

Use radical notation to write the expression. Simplify if possible. - $- 243 ^ { 1 / 5 }$

Raise to the power or find the root. Assume that all variables represent positive real numbers. Write with only positive exponents. - $\sqrt [ 3 ] { - 27 a ^ { 11 } b ^ { 13 } }$

Perform the indicated operation. Assume that all variables represent positive real numbers. - $( \sqrt { x } - 14 ) ^ { 2 }$

Write with positive exponents. Simplify if possible. - $\frac { 1 } { x ^ { - 8 / 3 } }$

Evaluate. -If $f ( x ) = \sqrt { 2 x - 7 }$ , find the value of $f ( 11 )$ .

Simplify. Assume that all variables represent positive real numbers. - $\frac { \sqrt { 168 } } { \sqrt { 6 } }$

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

Simplify. Assume that all variables represent any real number. - $\sqrt { 81 x ^ { 2 } }$

Find the cube root. - $\sqrt [ 3 ] { 512 x ^ { 6 } }$

Use the properties of exponents to simplify. Write with positive exponents. - $\frac { 12 ^ { 1 / 7 } } { 12 ^ { - 13 / 7 } }$

Add or subtract. Assume all variables represent positive real numbers. - $\frac { \sqrt [ 4 ] { 3750 } } { 11 x } - \frac { 2 \sqrt [ 4 ] { 6 } } { 22 x }$