Question 1

Which of the following correctly simplifies $$\displaystyle \frac{2x$$^{-3}$$y^2$$}{$$4x^2y$$^{-1}$$} to an $$expression with only positive exponents?

Accepted Answer

$$\displaystyle \frac{y^3$}{2x^5$}$$

Question 2

Evaluate $$\displaystyle 16$$^{\frac{3}$${4}}.$$

Accepted Answer

Question 3

Simplify $$\displaystyle (a$$^{\frac{1}$${2}}b$$^{\frac{1}$${3}})^6$ and express your answer with positive exponents.

Accepted Answer

$$\displaystyle a^3$b^2$

Question 4

Which of the following is equivalent to $$\displaystyle \sqrt[3]{x^4$y^5$$}$ in simplest radical form?

Accepted Answer

$$\displaystyle x\,y\sqrt[3]{xy^2$}$$

Question 5

Rationalize the denominator and simplify: $$\displaystyle \frac{5}{2\sqrt{3}}.$$

Accepted Answer

$$\displaystyle \frac{5\sqrt{3}}{6}$$

Question 6

Simplify $$\displaystyle \sqrt{50} + 2\sqrt{8} - \sqrt{18}.$$

Accepted Answer

$$\displaystyle 5\sqrt{2}$$

Question 7

Which of the following is the correct simplification of $$\displaystyle (2x$$^{\frac{1}$${2}}y$$^{\frac{1}$${3}}) \cdot (3x$$^{\frac{1}$${2}}y$$^{\frac{2}$${3}})$$?

Accepted Answer

$$\displaystyle 6x y$$

Question 8

Express $$\displaystyle \frac{1}{\sqrt[3]{4}} in $$simplest radical form with a rationalized denominator.

Accepted Answer

$$\displaystyle \frac{\sqrt[3]{16}}{4}$$

Question 9

Which of the following is a correct application of the exponent law $$a^m \cdot a^n = a$$^{m+n}$$?

Accepted Answer

$$\displaystyle x^2$ \cdot x$$^{-5}$$ = x$$^{-3}$$

Question 10

Simplify $$\displaystyle \frac{\sqrt{18} - \sqrt{8}}{\sqrt{2}}.$$

Accepted Answer

$$\displaystyle \sqrt{2}$$

Question 11

Which of the following is the correct simplification of $$\displaystyle (3 + \sqrt{5})(3 - \sqrt{5})$$?

Accepted Answer

$$\displaystyle 4$$

Question 12

A chemist needs to prepare a solution and must simplify $$\displaystyle \frac{\sqrt{12}}{\sqrt{3}} to $$determine the correct ratio. What is the simplified form?

Accepted Answer

$$\displaystyle 2$$

Question 13

Simplify $$\displaystyle \frac{2}{3 + \sqrt{5}} by $$rationalizing the denominator.

Accepted Answer

$$\displaystyle \frac{6 - 2\sqrt{5}}{4}$$

Question 14

Which of the following is the correct simplification of $$\displaystyle \sqrt[4]{81x^8$y^{12}$$}$?

Accepted Answer

$$\displaystyle 3x^2$y^3$

Question 15

A medical researcher is analyzing data and needs to simplify $$\displaystyle (16x^4$)^{\frac{3}$${4}}$. What is the result?

Accepted Answer

$$\displaystyle 32x^3$

Question 16

Which of the following is the correct simplification of $$\displaystyle \sqrt{32} + \sqrt{8}$$?

Accepted Answer

$$\displaystyle 6\sqrt{2}$$

Question 17

Simplify $$\displaystyle \frac{\sqrt{3} - 1}{\sqrt{3} + 1} by $$rationalizing the denominator.

Accepted Answer

$$\displaystyle \frac{2 - \sqrt{3}}{2}$$

Question 18

Which of the following is the correct simplification of $$\displaystyle (2\sqrt{3} + \sqrt{12}) - (\sqrt{27} - \sqrt{3})$$?

Accepted Answer

$$\displaystyle 2\sqrt{3}$$

Question 19

A physicist is working with the formula $$\displaystyle P = k\sqrt[3]{V}$$ and needs to express $$\displaystyle \sqrt[3]{54} in $$simplest radical form. What is the answer?

Accepted Answer

$$\displaystyle 3\sqrt[3]{2}$$

Chapter 11: Fractional Exponents and Radicals – Study Guide

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