Question 1

Simplify the expression. $125 ^ { \frac { 1 } { 3 } }$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $12$ 
C)  $\frac { 1 } { 28 }$ 
D)  $2$ 
E)  $5$ 
A)  $\frac { 1 } { 5 }$ 
B)  $12$ 
C)  $\frac { 1 } { 28 }$ 
D)  $2$ 
E)  $5$

Question 2

Perform the indicated operation and express your answer in simplest radical form. $\sqrt { 5 } \cdot \sqrt [ 3 ] { 5 }$

Accepted Answer

A)  $\sqrt [ 5 ] { 125 }$ 
B)  $\sqrt [ 5 ] { 3,125 }$ 
C)  $\sqrt [ 6 ] { 25 }$ 
D)  $\sqrt [ 6 ] { 5 }$ 
E)  $\sqrt [ 6 ] { 3,125 }$ 
A)  $\sqrt [ 5 ] { 125 }$ 
B)  $\sqrt [ 5 ] { 3,125 }$ 
C)  $\sqrt [ 6 ] { 25 }$ 
D)  $\sqrt [ 6 ] { 5 }$ 
E)  $\sqrt [ 6 ] { 3,125 }$

Question 3

Find the indicated product. Express the final result using positive integral exponents only. $$\left( 2 x y ^ { - 4 } \right) \left( 4 x ^ { - 4 } y ^ { 7 } \right)$$

Accepted Answer

A)  $\frac { 8 y ^ { 3 } } { x ^ { 3 } }$ 
B)  $8 x ^ { - 3 } y ^ { 3 }$ 
C)  $8 x ^ { 3 } y ^ { 3 }$ 
D)  $\frac { x ^ { 3 } } { 8 y ^ { 3 } }$ 
E)  $\frac { y ^ { 7 } } { 8 x y ^ { 8 } }$ 
A)  $\frac { 8 y ^ { 3 } } { x ^ { 3 } }$ 
B)  $8 x ^ { - 3 } y ^ { 3 }$ 
C)  $8 x ^ { 3 } y ^ { 3 }$ 
D)  $\frac { x ^ { 3 } } { 8 y ^ { 3 } }$ 
E)  $\frac { y ^ { 7 } } { 8 x y ^ { 8 } }$

Question 4

Solve the equation. Check all solutions. $\sqrt { x - 6 } - 4 = 5$

Accepted Answer

A) 85 
B) 64 
C) 87 
D) 76 
E) no solution 
A) 85 
B) 64 
C) 87 
D) 76 
E) no solution

Question 5

The distance between the Sun and Mercury is approximately $3.6 \times 10 ^ { 7 }$ miles. Use scientific notation to express this distance in feet. ( Hint: 5,280 feet = 1 mile.)

Accepted Answer

A)  $190.08 \times 10 ^ { 9 }$ ft 
B)  $0.19008 \times 10 ^ { 12 }$ ft 
C)  $1.9008 \times 10 ^ { 12 }$ ft 
D)  $1.9008 \times 10 ^ { 11 }$ ft 
E)  $19.008 \times 10 ^ { 10 }$ ft 
A)  $190.08 \times 10 ^ { 9 }$ ft 
B)  $0.19008 \times 10 ^ { 12 }$ ft 
C)  $1.9008 \times 10 ^ { 12 }$ ft 
D)  $1.9008 \times 10 ^ { 11 }$ ft 
E)  $19.008 \times 10 ^ { 10 }$ ft

Question 6

Write the number $511 \times 10 ^ { - 2 }$ in standard notation.

Accepted Answer

A) 5.11 
B) 51.1 
C) 5,110 
D) 0.00511 
E) 51,100 
A) 5.11 
B) 51.1 
C) 5,110 
D) 0.00511 
E) 51,100

Question 7

Find a rational approximation, to the nearest tenth, for the radical expression. $7 \sqrt { 3 } - 8 \sqrt { 3 } + 10 \sqrt { 3 } - 7 \sqrt { 3 } - 9 \sqrt { 3 }$

Accepted Answer

A)  $- 8.8$ 
B)  $- 11.4$ 
C)  $- 13.1$ 
D)  $- 11.3$ 
E)  $- 12.1$ 
A)  $- 8.8$ 
B)  $- 11.4$ 
C)  $- 13.1$ 
D)  $- 11.3$ 
E)  $- 12.1$

Question 8

Rationalize the denominator and simplify. $\frac { 21 } { \sqrt { 21 } + 1 }$

Accepted Answer

A)  $\frac { 21 ( \sqrt { 21 } - 1 ) } { 20 }$ 
B)  $\frac { 21 \sqrt { 21 } - 1 } { 20 }$ 
C)  $\frac { 20 ( \sqrt { 21 } - 1 ) } { 21 }$ 
D)  $\frac { \sqrt { 21 } - 21 } { 20 }$ 
E)  $21 ( \sqrt { 21 } - 1 )$ 
A)  $\frac { 21 ( \sqrt { 21 } - 1 ) } { 20 }$ 
B)  $\frac { 21 \sqrt { 21 } - 1 } { 20 }$ 
C)  $\frac { 20 ( \sqrt { 21 } - 1 ) } { 21 }$ 
D)  $\frac { \sqrt { 21 } - 21 } { 20 }$ 
E)  $21 ( \sqrt { 21 } - 1 )$

Question 9

Change the radical to an exponential expression. $\sqrt { m ^ { 2 } + k ^ { 2 } }$

Accepted Answer

A)  $\left( m ^ { 2 } + k ^ { 2 } \right) ^ { 2 }$ 
B)  $m ^ { \frac { 3 } { 2 } } + k ^ { \frac { 3 } { 2 } }$ 
C)  $( m + k ) ^ { \frac { 1 } { 2 } }$ 
D)  $\left( m ^ { 2 } + k ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ 
E)  $( m + k ) ^ { \frac { 3 } { 2 } }$ 
A)  $\left( m ^ { 2 } + k ^ { 2 } \right) ^ { 2 }$ 
B)  $m ^ { \frac { 3 } { 2 } } + k ^ { \frac { 3 } { 2 } }$ 
C)  $( m + k ) ^ { \frac { 1 } { 2 } }$ 
D)  $\left( m ^ { 2 } + k ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ 
E)  $( m + k ) ^ { \frac { 3 } { 2 } }$

Question 10

Simplify the expression. Express final result without using zero or negative integers as exponents. $$\left( \frac { x ^ { - 1 } } { y ^ { - 4 } } \right) ^ { - 3 }$$

Accepted Answer

A)  $\frac { x ^ { 3 } } { y ^ { 12 } }$ 
B)  $\frac { y ^ { 7 } } { x ^ { 4 } }$ 
C)  $\frac { x ^ { 4 } } { y ^ { 12 } }$ 
D)  $\frac { y ^ { 12 } } { x ^ { 3 } }$ 
E)  $\frac { x ^ { 4 } } { y ^ { 7 } }$ 
A)  $\frac { x ^ { 3 } } { y ^ { 12 } }$ 
B)  $\frac { y ^ { 7 } } { x ^ { 4 } }$ 
C)  $\frac { x ^ { 4 } } { y ^ { 12 } }$ 
D)  $\frac { y ^ { 12 } } { x ^ { 3 } }$ 
E)  $\frac { x ^ { 4 } } { y ^ { 7 } }$

Question 11

Find the indicated product. Express the final result using positive integral exponents only. $$3 x y ^ { - 2 } 5 x ^ { - 3 } y ^ { 4 }$$

Accepted Answer

A)  $\frac { y ^ { 4 } } { 15 x y ^ { 5 } }$ 
B)  $15 x ^ { - 2 } y ^ { 2 }$ 
C)  $\frac { x ^ { 2 } } { 15 y ^ { 2 } }$ 
D)  $15 x ^ { 2 } y ^ { 2 }$ 
E)  $\frac { 15 y ^ { 2 } } { x ^ { 2 } }$ 
A)  $\frac { y ^ { 4 } } { 15 x y ^ { 5 } }$ 
B)  $15 x ^ { - 2 } y ^ { 2 }$ 
C)  $\frac { x ^ { 2 } } { 15 y ^ { 2 } }$ 
D)  $15 x ^ { 2 } y ^ { 2 }$ 
E)  $\frac { 15 y ^ { 2 } } { x ^ { 2 } }$

Question 12

Write $x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 4 } }$ in radical form. For example, $3 x ^ { \frac { 2 } { 3 } } = 3 \sqrt [ 3 ] { x ^ { 2 } }$ .

Accepted Answer

A)  $\frac { 1 } { \sqrt [ 4 ] { x ^ { 3 } y } }$ 
B)  $\sqrt [ 4 ] { x ^ { 3 } y }$ 
C)  $\sqrt { x ^ { 4 } y }$ 
D)  $y \sqrt [ 4 ] { x ^ { 3 } }$ 
E)  $\sqrt [ 3 ] { x ^ { 4 } y }$ 
A)  $\frac { 1 } { \sqrt [ 4 ] { x ^ { 3 } y } }$ 
B)  $\sqrt [ 4 ] { x ^ { 3 } y }$ 
C)  $\sqrt { x ^ { 4 } y }$ 
D)  $y \sqrt [ 4 ] { x ^ { 3 } }$ 
E)  $\sqrt [ 3 ] { x ^ { 4 } y }$

Question 13

Simplify the expression. $27 ^ { \frac { 1 } { 3 } }$

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $9$ 
C)  $\frac { 1 } { 32 }$ 
D)  $2$ 
E)  $3$ 
A)  $\frac { 1 } { 3 }$ 
B)  $9$ 
C)  $\frac { 1 } { 32 }$ 
D)  $2$ 
E)  $3$

Question 14

Rationalize the denominator and simplify. The variable represents a positive real number. $\frac { \sqrt { x } } { \sqrt { x } - 10 }$

Accepted Answer

A)  $\frac { 10 \sqrt { x } } { x - 100 }$ 
B)  $\frac { x + 10 \sqrt { x } } { x - 10 }$ 
C)  $\sqrt { x }$ 
D)  $\frac { 11 x } { x - 100 }$ 
E)  $\frac { x + 10 \sqrt { x } } { x - 100 }$ 
A)  $\frac { 10 \sqrt { x } } { x - 100 }$ 
B)  $\frac { x + 10 \sqrt { x } } { x - 10 }$ 
C)  $\sqrt { x }$ 
D)  $\frac { 11 x } { x - 100 }$ 
E)  $\frac { x + 10 \sqrt { x } } { x - 100 }$

Question 15

Johannes Kepler discovered that a planet's mean distance $R$ from the sun (in astronomical units)is related to its period $T$ (in years)by the formula $R = \sqrt [ 3 ] { \frac { T ^ { 2 } } { k } }$ Find $R$ when $T = 1.881$ and $k = 1.002$ .

Accepted Answer

A) 1.52 AU 
B) 1.22 AU 
C) 1.37 AU 
D) 1.35 AU 
E) 1.45 AU 
A) 1.52 AU 
B) 1.22 AU 
C) 1.37 AU 
D) 1.35 AU 
E) 1.45 AU

Question 16

Write the quotient as the quotient of two radicals and simplify. $\sqrt { \frac { 169 } { 100 } }$

Accepted Answer

A)  $\frac { 7 } { 10 }$ 
B)  $\frac { 10 } { 7 }$ 
C)  $\frac { 13 } { 10 }$ 
D)  $\frac { 10 } { 13 }$ 
E)  $\frac { 169 } { 10 }$ 
A)  $\frac { 7 } { 10 }$ 
B)  $\frac { 10 } { 7 }$ 
C)  $\frac { 13 } { 10 }$ 
D)  $\frac { 10 } { 13 }$ 
E)  $\frac { 169 } { 10 }$

Question 17

Use the distributive property to help simplify the expression. $2 \sqrt [ 3 ] { 2 } + 2 \sqrt [ 3 ] { 54 } - 2 \sqrt [ 3 ] { 16 }$

Accepted Answer

A)  $6 \sqrt [ 3 ] { 2 }$ 
B)  $- 4 \sqrt [ 3 ] { 2 }$ 
C)  $- 4 \sqrt [ 3 ] { 3 }$ 
D)  $4 \sqrt [ 3 ] { 2 }$ 
E)  $4 \sqrt [ 3 ] { 3 }$ 
A)  $6 \sqrt [ 3 ] { 2 }$ 
B)  $- 4 \sqrt [ 3 ] { 2 }$ 
C)  $- 4 \sqrt [ 3 ] { 3 }$ 
D)  $4 \sqrt [ 3 ] { 2 }$ 
E)  $4 \sqrt [ 3 ] { 3 }$

Question 18

Write the number 330,000 in scientific notation.

Accepted Answer

A)  $33 \times 10 ^ { 5 }$ 
B)  $33 \times 10 ^ { 6 }$ 
C)  $3.3 \times 10 ^ { 4 }$ 
D)  $33 \times 10 ^ { 4 }$ 
E)  $3.3 \times 10 ^ { 5 }$ 
A)  $33 \times 10 ^ { 5 }$ 
B)  $33 \times 10 ^ { 6 }$ 
C)  $3.3 \times 10 ^ { 4 }$ 
D)  $33 \times 10 ^ { 4 }$ 
E)  $3.3 \times 10 ^ { 5 }$

Question 19

The formula $T = 2 \pi \sqrt { \frac { L } { 32 } }$ represents the period of a pendulum T (in seconds), L - the length of the pendulum (in feet). Solve the formula for L if T = 39.4384 sec, π = 3.14.

Accepted Answer

A) 78.8768 feet 
B) 100.48 feet 
C) 128 feet 
D) 32 feet 
A) 78.8768 feet 
B) 100.48 feet 
C) 128 feet 
D) 32 feet

Question 20

The number of wrenches that can be produced at a given price can be predicted by the formula $s = \sqrt { 65 x }$ , where $s$ is the supply (in thousands)and $X$ is the price (in dollars). If the demand $d$ for the wrenches can be predicted by the formula $d = \sqrt { 324 - 4 x ^ { 2 } }$ , find the equilibrium price.

Accepted Answer

A) $6 
B) $5 
C) $4 
D) None of the above 
A) $6 
B) $5 
C) $4 
D) None of the above

Simplify the expression. $125 ^ { \frac { 1 } { 3 } }$

Perform the indicated operation and express your answer in simplest radical form. $\sqrt { 5 } \cdot \sqrt [ 3 ] { 5 }$

Find the indicated product. Express the final result using positive integral exponents only. $\left( 2 x y ^ { - 4 } \right) \left( 4 x ^ { - 4 } y ^ { 7 } \right)$

Solve the equation. Check all solutions. $\sqrt { x - 6 } - 4 = 5$

The distance between the Sun and Mercury is approximately $3.6 \times 10 ^ { 7 }$ miles. Use scientific notation to express this distance in feet. ( Hint: 5,280 feet = 1 mile.)

Write the number $511 \times 10 ^ { - 2 }$ in standard notation.

Find a rational approximation, to the nearest tenth, for the radical expression. $7 \sqrt { 3 } - 8 \sqrt { 3 } + 10 \sqrt { 3 } - 7 \sqrt { 3 } - 9 \sqrt { 3 }$

Rationalize the denominator and simplify. $\frac { 21 } { \sqrt { 21 } + 1 }$

Change the radical to an exponential expression. $\sqrt { m ^ { 2 } + k ^ { 2 } }$

Simplify the expression. Express final result without using zero or negative integers as exponents. $\left( \frac { x ^ { - 1 } } { y ^ { - 4 } } \right) ^ { - 3 }$

Find the indicated product. Express the final result using positive integral exponents only. $3 x y ^ { - 2 } 5 x ^ { - 3 } y ^ { 4 }$

Write $x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 4 } }$ in radical form. For example, $3 x ^ { \frac { 2 } { 3 } } = 3 \sqrt [ 3 ] { x ^ { 2 } }$ .

Simplify the expression. $27 ^ { \frac { 1 } { 3 } }$

Rationalize the denominator and simplify. The variable represents a positive real number. $\frac { \sqrt { x } } { \sqrt { x } - 10 }$

Johannes Kepler discovered that a planet's mean distance $R$ from the sun (in astronomical units)is related to its period $T$ (in years)by the formula $R = \sqrt [ 3 ] { \frac { T ^ { 2 } } { k } }$ Find $R$ when $T = 1.881$ and $k = 1.002$ .

Write the quotient as the quotient of two radicals and simplify. $\sqrt { \frac { 169 } { 100 } }$

Use the distributive property to help simplify the expression. $2 \sqrt [ 3 ] { 2 } + 2 \sqrt [ 3 ] { 54 } - 2 \sqrt [ 3 ] { 16 }$

Write the number 330,000 in scientific notation.

The formula $T = 2 \pi \sqrt { \frac { L } { 32 } }$ represents the period of a pendulum T (in seconds), L - the length of the pendulum (in feet). Solve the formula for L if T = 39.4384 sec, π = 3.14.

Basic Concepts and Properties

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Exam 5: Exponents and Radicals

Simplify the expression. 12513125 ^ { \frac { 1 } { 3 } }12531​

Perform the indicated operation and express your answer in simplest radical form. 5⋅53\sqrt { 5 } \cdot \sqrt [ 3 ] { 5 }5​⋅35​

Find the indicated product. Express the final result using positive integral exponents only. (2xy−4)(4x−4y7)\left( 2 x y ^ { - 4 } \right) \left( 4 x ^ { - 4 } y ^ { 7 } \right)(2xy−4)(4x−4y7)

Solve the equation. Check all solutions. x−6−4=5\sqrt { x - 6 } - 4 = 5x−6​−4=5

The distance between the Sun and Mercury is approximately 3.6×1073.6 \times 10 ^ { 7 }3.6×107 miles. Use scientific notation to express this distance in feet. ( Hint: 5,280 feet = 1 mile.)

Write the number 511×10−2511 \times 10 ^ { - 2 }511×10−2 in standard notation.

Find a rational approximation, to the nearest tenth, for the radical expression. 73−83+103−73−937 \sqrt { 3 } - 8 \sqrt { 3 } + 10 \sqrt { 3 } - 7 \sqrt { 3 } - 9 \sqrt { 3 }73​−83​+103​−73​−93​

Rationalize the denominator and simplify. 2121+1\frac { 21 } { \sqrt { 21 } + 1 }21​+121​

Change the radical to an exponential expression. m2+k2\sqrt { m ^ { 2 } + k ^ { 2 } }m2+k2​

Simplify the expression. Express final result without using zero or negative integers as exponents. (x−1y−4)−3\left( \frac { x ^ { - 1 } } { y ^ { - 4 } } \right) ^ { - 3 }(y−4x−1​)−3

Find the indicated product. Express the final result using positive integral exponents only. 3xy−25x−3y43 x y ^ { - 2 } 5 x ^ { - 3 } y ^ { 4 }3xy−25x−3y4

Write x34y14x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 4 } }x43​y41​ in radical form. For example, 3x23=3x233 x ^ { \frac { 2 } { 3 } } = 3 \sqrt [ 3 ] { x ^ { 2 } }3x32​=33x2​ .

Simplify the expression. 271327 ^ { \frac { 1 } { 3 } }2731​

Rationalize the denominator and simplify. The variable represents a positive real number. xx−10\frac { \sqrt { x } } { \sqrt { x } - 10 }x​−10x​​

Johannes Kepler discovered that a planet's mean distance RRR from the sun (in astronomical units)is related to its period TTT (in years)by the formula R=T2k3R = \sqrt [ 3 ] { \frac { T ^ { 2 } } { k } }R=3kT2​​ Find RRR when T=1.881T = 1.881T=1.881 and k=1.002k = 1.002k=1.002 .

Write the quotient as the quotient of two radicals and simplify. 169100\sqrt { \frac { 169 } { 100 } }100169​​

Use the distributive property to help simplify the expression. 223+2543−21632 \sqrt [ 3 ] { 2 } + 2 \sqrt [ 3 ] { 54 } - 2 \sqrt [ 3 ] { 16 }232​+2354​−2316​

Write the number 330,000 in scientific notation.

The formula T=2πL32T = 2 \pi \sqrt { \frac { L } { 32 } }T=2π32L​​ represents the period of a pendulum T (in seconds), L - the length of the pendulum (in feet). Solve the formula for L if T = 39.4384 sec, π = 3.14.

Basic Concepts and Properties

Equations, Inequalities, and Problem Solving

Polynomials

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Filters

Simplify the expression. $125 ^ { \frac { 1 } { 3 } }$

Perform the indicated operation and express your answer in simplest radical form. $\sqrt { 5 } \cdot \sqrt [ 3 ] { 5 }$

Find the indicated product. Express the final result using positive integral exponents only. $\left( 2 x y ^ { - 4 } \right) \left( 4 x ^ { - 4 } y ^ { 7 } \right)$

Solve the equation. Check all solutions. $\sqrt { x - 6 } - 4 = 5$

The distance between the Sun and Mercury is approximately $3.6 \times 10 ^ { 7 }$ miles. Use scientific notation to express this distance in feet. ( Hint: 5,280 feet = 1 mile.)

Write the number $511 \times 10 ^ { - 2 }$ in standard notation.

Find a rational approximation, to the nearest tenth, for the radical expression. $7 \sqrt { 3 } - 8 \sqrt { 3 } + 10 \sqrt { 3 } - 7 \sqrt { 3 } - 9 \sqrt { 3 }$

Rationalize the denominator and simplify. $\frac { 21 } { \sqrt { 21 } + 1 }$

Change the radical to an exponential expression. $\sqrt { m ^ { 2 } + k ^ { 2 } }$

Simplify the expression. Express final result without using zero or negative integers as exponents. $\left( \frac { x ^ { - 1 } } { y ^ { - 4 } } \right) ^ { - 3 }$

Find the indicated product. Express the final result using positive integral exponents only. $3 x y ^ { - 2 } 5 x ^ { - 3 } y ^ { 4 }$

Write $x ^ { \frac { 3 } { 4 } } y ^ { \frac { 1 } { 4 } }$ in radical form. For example, $3 x ^ { \frac { 2 } { 3 } } = 3 \sqrt [ 3 ] { x ^ { 2 } }$ .

Simplify the expression. $27 ^ { \frac { 1 } { 3 } }$

Rationalize the denominator and simplify. The variable represents a positive real number. $\frac { \sqrt { x } } { \sqrt { x } - 10 }$

Johannes Kepler discovered that a planet's mean distance $R$ from the sun (in astronomical units)is related to its period $T$ (in years)by the formula $R = \sqrt [ 3 ] { \frac { T ^ { 2 } } { k } }$ Find $R$ when $T = 1.881$ and $k = 1.002$ .

Write the quotient as the quotient of two radicals and simplify. $\sqrt { \frac { 169 } { 100 } }$

Use the distributive property to help simplify the expression. $2 \sqrt [ 3 ] { 2 } + 2 \sqrt [ 3 ] { 54 } - 2 \sqrt [ 3 ] { 16 }$

The formula $T = 2 \pi \sqrt { \frac { L } { 32 } }$ represents the period of a pendulum T (in seconds), L - the length of the pendulum (in feet). Solve the formula for L if T = 39.4384 sec, π = 3.14.