In some cases, rational exponents (fractional exponents) allow us to work with radicals more easily. Most commonly, this allows us to write the nth root of a number or expression as being raised to the power of 1/n.

Test Objectives
• Demonstrate the ability to convert from radical notation to exponential notation
• Demonstrate the ability to convert from exponential notation to radical notation
• Demonstrate a general understanding of the rules of exponents
Rational Exponents Practice Test:

#1:

Instructions: Rewrite in exponential form.

$$a)\hspace{.2em}\sqrt{x - 7}$$

$$b)\hspace{.2em}\sqrt[3]{(x + 1)^2}$$

#2:

Instructions: Rewrite in exponential form.

$$a)\hspace{.2em}\sqrt[3]{(x^2 - 3)}$$

$$b)\hspace{.2em}(\sqrt[3]{7x})^4$$

#3:

$$a)\hspace{.2em}(6x)^{\frac{1}{2}}$$

$$b)\hspace{.2em}\frac{1}{(3x - 5)^{-\frac{3}{2}}}$$

#4:

Instructions: Simplify each, write in radical form.

$$a)\hspace{.2em}\frac{(-4x^2 - 5)^{\frac{1}{2}}}{x^2}$$

$$b)\hspace{.2em}\frac{x^{\frac{4}{3}}y^{\frac{7}{3}}z^{-2}}{(xy)^{\frac{1}{3}}}$$

#5:

Instructions: Simplify each, write in radical form.

$$a)\hspace{.2em}\frac{x^{\frac{2}{3}}y^{\frac{5}{3}}z^{\frac{1}{3}}}{(x^{\frac{1}{5}}y^{-\frac{5}{2}}z^{-\frac{2}{3}})^0}$$

$$b)\hspace{.2em}(y^{-\frac{3}{5}}z^{\frac{4}{3}})^{-\frac{1}{2}}\cdot (x^{\frac{1}{4}}y^{\frac{1}{2}}z^{\frac{3}{2}})^2$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(x-7)^{\frac{1}{2}}$$

$$b)\hspace{.2em}(x + 1)^{\frac{2}{3}}$$

#2:

Solutions:

$$a)\hspace{.2em}(x^2 - 3)^{\frac{1}{3}}$$

$$b)\hspace{.2em}7^{\frac{4}{3}}x^{\frac{4}{3}}$$

#3:

Solutions:

$$a)\hspace{.2em}\sqrt{6x}$$

$$b)\hspace{.2em}(\sqrt{3x - 5})^3$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{\sqrt{-4x^2-5}}{x^2}$$

$$b)\hspace{.2em}\frac{xy^2}{z^2}$$

#5:

Solutions:

$$a)\hspace{.2em}y\sqrt[3]{x^2y^2z}$$

$$b)\hspace{.2em}yz^2 \sqrt[10]{y^3}\sqrt[3]{z}\sqrt{x}$$