Question

which expression is equivalent to \\(\frac{\sqrt{10}}{\sqrt4{8}}\\)?

• \\(\frac{\sqrt4{200}}{2}\\)
• \\(\frac{\sqrt4{20}}{2}\\)
• \\(\frac{2\sqrt{5}}{5}\\)
• \\(\frac{100}{8}\\)

Explanation:

Convert to a common index

Using the Common Index Radicals knowledge point

$$ \frac{\sqrt{10}}{\sqrt[4]{8}} = \frac{10^{\frac{1}{2}}}{8^{\frac{1}{4}}} = \frac{10^{\frac{2}{4}}}{8^{\frac{1}{4}}} = \frac{\sqrt[4]{10^2}}{\sqrt[4]{8}} = \frac{\sqrt[4]{100}}{\sqrt[4]{8}} $$

Simplify the radical quotient

Using the Simplifying Radical Quotients knowledge point

$$ \frac{\sqrt[4]{100}}{\sqrt[4]{8}} = \sqrt[4]{\frac{100}{8}} = \sqrt[4]{\frac{25}{2}} $$

Rationalize the denominator

Using the Rationalizing the Denominator knowledge point

$$ \sqrt[4]{\frac{25}{2}} = \frac{\sqrt[4]{25}}{\sqrt[4]{2}} \cdot \frac{\sqrt[4]{8}}{\sqrt[4]{8}} = \frac{\sqrt[4]{200}}{\sqrt[4]{16}} = \frac{\sqrt[4]{200}}{2} $$

Answer:

• (A) \(\frac{\sqrt[4]{200}}{2}\) (Correct answer)
• (B) \(\frac{\sqrt[4]{200}}{2}\) is incorrect, this option is \(\frac{\sqrt[4]{200}}{2}\) but let's list the others: \(\frac{\sqrt[4]{20}}{2}\)
• (C) \(\frac{2\sqrt{5}}{5}\)
• (D) \(\frac{100}{8}\)