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When simplifying expressions involving square roots, you might encounter a situation where there is a square root in the denominator. In mathematics, it is often desirable to remove, or 'rationalize' the square root from the denominator. This process is called 'rationalizing the denominator'.
Doing so helps in making the fractions easier to understand and work with.

Here's how we can rationalize the denominator of \(\frac{\sqrt{2}}{\sqrt{7}}\).
To eliminate the square root from the denominator, we multiply both the numerator and the denominator by the same square root found in the denominator. In this case, we multiply both by \(\sqrt{7}\).

This results in:
\(\frac{\sqrt{2} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} \).
The denominator becomes \(\ \sqrt{7} \times \sqrt{7} = 7 \. \).
Thus, the expression simplifies to \(\frac{\sqrt{14}}{7}\).

Understanding the properties of square roots is crucial when dealing with expressions under a radical sign.
One such property states that the square root of a fraction \( \ \frac{a}{b} = \frac{\sqrt{a}}{\sqrt{b}} \. \)
This means you can separate the numerator and the denominator into their own square roots.

So, for the given expression \(\ \sqrt{\frac{2}{7}} \, \), we can rewrite it as:

\(\ \frac{\sqrt{2}}{\sqrt{7}} \. \)
This property is particularly useful in simplifying complex expressions involving square roots.
It allows us to handle the numerator and denominator individually and then perform further simplification as needed.
Remembering and applying this property is an essential step in solving square root problems.

When finding the square root of a fraction, the key idea is to split the fraction into two separate square root terms—the numerator and the denominator.
This involves using one of the fundamental properties of square roots:

\( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Take the given fraction under the square root: \(\ \sqrt{\frac{2}{7}} \. \)
Using the property mentioned above, we rewrite it as:

\(\ \frac{\sqrt{2}}{\sqrt{7}} \. \)

Now, the fraction no longer has a single square root operation, but two separate ones. This separation allows for further adjustments and simplifications.

Therefore, you follow the steps of rationalizing the denominator (if necessary), to the final simplified form. Applying these basic but powerful principles helps you systematically tackle square root problems involving fractions.