Question

the following work models how to write the quotient \\(\frac{\sqrt3{4}}{\sqrt{2}}\\) using rational exponents.

\\\frac{\sqrt3{4}}{\sqrt{2}} = \frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}} = \frac{\left(2^2\
ight)^{\frac{1}{3}}}{2^{\frac{1}{2}}} = \frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}\\

note that, in order to use the quotient property of exponents, the bases must be the same.

using the work shown on the left, what is \\(\frac{\sqrt3{4}}{\sqrt{2}}\\) in simplest radical form?

\\(\bigcirc\\) 2
\\(\bigcirc\\) \\(\sqrt{2^6}\\)
\\(\bigcirc\\) \\(\sqrt6{2}\\)

Explanation:

Convert to rational exponents

Using the Radical to Exponential Form knowledge point

$$ \frac{\sqrt[3]{4}}{\sqrt{2}} = \frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}} $$

Apply quotient property of exponents

Using the Rational Exponents knowledge point

$$ 2^{\frac{2}{3} - \frac{1}{2}} = 2^{\frac{4}{6} - \frac{3}{6}} = 2^{\frac{1}{6}} $$

Convert back to radical form

Using the Radical to Exponential Form knowledge point

$$ 2^{\frac{1}{6}} = \sqrt[6]{2} $$

Answer:

• 2
• \(\sqrt{2^6}\)
• \(\sqrt[6]{2}\) (Correct answer)