To simplify the expression \(\frac{\sqrt{3}}{\sqrt{2} - \sqrt{5}}\), we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(\sqrt{2} - \sqrt{5}\) is \(\sqrt{2} + \sqrt{5}\).

Here's the process:

\[
\frac{\sqrt{3}}{\sqrt{2} - \sqrt{5}} \times \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}} = \frac{\sqrt{3}(\sqrt{2} + \sqrt{5})}{(\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5})}
\]

Now, simplify the denominator using the difference of squares formula \((a - b)(a + b) = a^2 - b^2\):

\[
(\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5}) = (\sqrt{2})^2 - (\sqrt{5})^2 = 2 - 5 = -3
\]

So, the expression becomes:

\[
\frac{\sqrt{3}(\sqrt{2} + \sqrt{5})}{-3}
\]

Distribute \(\sqrt{3}\) in the numerator:

\[
\frac{\sqrt{3}\sqrt{2} + \sqrt{3}\sqrt{5}}{-3}
\]

This can be written as:

\[
\frac{\sqrt{6} + \sqrt{15}}{-3}
\]

Finally, we can write this as:

\[
-\frac{\sqrt{6} + \sqrt{15}}{3}
\]

Thus, the simplified form of the expression \(\frac{\sqrt{3}}{\sqrt{2} - \sqrt{5}}\) is:

\[
-\frac{\sqrt{6} + \sqrt{15}}{3}
\]

Would you like more details or have any questions? Here are some related problems:

1. Simplify \(\frac{\sqrt{7}}{\sqrt{3} - \sqrt{2}}\).
2. Rationalize the denominator of \(\frac{\sqrt{5}}{\sqrt{3} + \sqrt{7}}\).
3. Simplify \(\frac{\sqrt{8}}{\sqrt{6} - \sqrt{2}}\).
4. Rationalize \(\frac{1}{\sqrt{2} + \sqrt{5}}\).
5. Simplify \(\frac{\sqrt{10}}{\sqrt{5} - \sqrt{3}}\).

**Tip:** Always check your simplified expression by multiplying it back out to ensure it matches the original expression.

Learn how to simplify the expression √3/√2 - √5 with a detailed step-by-step solution. This problem involves rationalizing the denominator using the difference of squares formula. Suitable for high school students.

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