Question 1

Which of the following expressions is equivalent to $$\sqrt[3]{x^4$}$$ for $$x \u003e 0$$?

Accepted Answer

$$x^{4/3}$$

Question 2

Simplify the expression $$\$$left(16x^8$$\$$right)^{1/4}$$ for x$$ \u003e 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 $$a \u003e 0.$$

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$$ \u003e 0$$.

Accepted Answer

$$x^2$$

Question 8

Which of the following is equivalent to $$\dfrac{1}{x$$^{-2/5}$$}$$ for $$x \u003e 0$$?

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}$$ for x$$ \u003e 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 $$x \u003e 0$$?

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$$, b \u003e 0$$.

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 $$x \u003e 0$$?

Accepted Answer

$$x^{3/2}$$

Question 18

Simplify $$\$$left(x^{-1/3}$$y^{2/3}$$\$$right)^3$$ for $$x, y \u003e 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

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