Question 1

Which of the following expressions is equivalent to $\sqrt[3]{x^4}$ for $x > 0$?

Accepted Answer

$x^{4/3}$

Question 2

Simplify the expression $\left(16x^8\right)^{1/4}$ for $x > 0$.

Accepted Answer

$2x^2$

Question 3

Which of the following is NOT a real number?

Accepted Answer

$\sqrt{-16}$

Question 4

Rewrite $\dfrac{a^{2/3}}{a^{5/6}}$ with positive exponents only, for $a > 0$.

Accepted Answer

$\dfrac{1}{a^{1/6}}$

Question 5

Evaluate $81^{3/4}$.

Accepted Answer

$27$

Question 6

A chemist needs to prepare a solution and must take the cube root of the volume $V = 125$ mL. What is $V^{1/3}$?

Accepted Answer

$5$

Question 7

Simplify the expression $\left(\dfrac{x^{1/2}}{x^{1/4}}\right)^4$ for $x > 0$.

Accepted Answer

$x^2$

Question 8

Which of the following is equivalent to $\dfrac{1}{x^{-2/5}}$ for $x > 0$?

Accepted Answer

$x^{2/5}$

Question 9

A medical researcher is analyzing the growth of bacteria, which doubles every hour. If the initial population is $P_0$, what is the population after $t$ hours, expressed using exponents?

Accepted Answer

$P_0 \cdot 2^t$

Question 10

Simplify $\left(27x^6\right)^{2/3}$ for $x > 0$.

Accepted Answer

$9x^4$

Question 11

Which of the following is the correct application of the negative exponent rule?

Accepted Answer

$a^{-3} = \dfrac{1}{a^3}$

Question 12

Evaluate $16^{-3/4}$.

Accepted Answer

$\dfrac{1}{8}$

Question 13

Which of the following is equivalent to $\sqrt[5]{x^3}$ for $x > 0$?

Accepted Answer

$x^{3/5}$

Question 14

Simplify $\left(\dfrac{a^{1/2}b^{1/3}}{a^{1/6}b^{2/3}}\right)^6$ for $a, b > 0$.

Accepted Answer

$a^3b^{-2}$

Question 15

A historian is analyzing population growth in a city, modeled by $P = 5000 \cdot (1.02)^t$, where $t$ is the number of years since 2000. What is the population in 10 years?

Accepted Answer

$5000 \cdot (1.02)^{10}$

Question 16

Evaluate $\left(\dfrac{27}{8}\right)^{-2/3}$.

Accepted Answer

$\dfrac{4}{9}$

Question 17

Which of the following is equivalent to $\sqrt[4]{x^6}$ for $x > 0$?

Accepted Answer

$x^{3/2}$

Question 18

Simplify $\left(x^{-1/3}y^{2/3}\right)^3$ for $x, y > 0$.

Accepted Answer

$\dfrac{y^2}{x}$

Question 19

Evaluate $(-64)^{1/3}$.

Accepted Answer

$-4$

Question 20

Which of the following is the correct application of the product rule for exponents?

Accepted Answer

$a^{m} \cdot a^{n} = a^{m+n}$

Radicals and Rational Exponents: Properties, Simplification, and Evaluation

Study Guide - Practice Questions

Study Guide - Flashcards