To simplify the expression \(\frac{\sqrt{2}}{\sqrt{5} - 1}\), we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(\sqrt{5} - 1\) is \(\sqrt{5} + 1\).

Here's the process:

\[
\frac{\sqrt{2}}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1} = \frac{\sqrt{2}(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)}
\]

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

\[
(\sqrt{5} - 1)(\sqrt{5} + 1) = (\sqrt{5})^2 - (1)^2 = 5 - 1 = 4
\]

So, the expression becomes:

\[
\frac{\sqrt{2}(\sqrt{5} + 1)}{4}
\]

Distribute \(\sqrt{2}\) in the numerator:

\[
\frac{\sqrt{2}\sqrt{5} + \sqrt{2}}{4}
\]

Which can be written as:

\[
\frac{\sqrt{10} + \sqrt{2}}{4}
\]

Thus, the simplified form of the expression \(\frac{\sqrt{2}}{\sqrt{5} - 1}\) is:

\[
\frac{\sqrt{10} + \sqrt{2}}{4}
\]

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

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

**Tip:** Rationalizing the denominator helps eliminate radicals from the denominator, making the expression easier to work with in subsequent calculations.

Learn how to simplify the expression √2/√5 - 1 using the rationalization technique. Step-by-step solution provided, covering radicals and the difference of squares formula. Suitable for high school students.

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