Question

simplify the radical expression by rationalizing the denominator.

\\\sqrt{\frac{36}{10}}\\

give your answer in simplest form.

Explanation:

Apply quotient property of radicals

We split the radical of the fraction into separate radicals for the numerator and denominator.

$$ \sqrt{\frac{36}{10}} = \frac{\sqrt{36}}{\sqrt{10}} $$

Simplify the numerator

We evaluate the square root of the perfect square in the numerator.

$$ \sqrt{36} = 6 $$

This simplifies our expression to:

$$ \frac{6}{\sqrt{10}} $$

Rationalize the denominator

To eliminate the radical in the denominator, we multiply both the numerator and the denominator by \(\sqrt{10}\).

$$ \frac{6}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{6\sqrt{10}}{10} $$

Simplify the fraction

We reduce the fraction by dividing the numerator and the denominator by their greatest common divisor, which is \(2\).

$$ \frac{6\sqrt{10}}{10} = \frac{3\sqrt{10}}{5} $$

Answer:

Simplify the radical expression by rationalizing the denominator.

$$\sqrt{\frac{36}{10}}$$

The simplified expression is <blank>\(\frac{3\sqrt{10}}{5}\)</blank>.