Question

simplify the radical expression by rationalizing the denominator.

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

give your answer in simplest form.

Explanation:

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<reasoning>

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 \(2\).
\[
\frac{6\sqrt{10}}{10} = \frac{3\sqrt{10}}{5}
\]
</reasoning>

<answer>
Simplify the radical expression by rationalizing the denominator.

\[\sqrt{\frac{36}{10}}\]

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

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"Mathematics",
"Algebra",
"Rationalizing Denominators"
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Answer:

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<reasoning>

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 \(2\).
\[
\frac{6\sqrt{10}}{10} = \frac{3\sqrt{10}}{5}
\]
</reasoning>

<answer>
Simplify the radical expression by rationalizing the denominator.

\[\sqrt{\frac{36}{10}}\]

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

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"Mathematics",
"Algebra",
"Rationalizing Denominators"
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