To simplify the expression \(\frac{\sqrt{8}}{1 - \sqrt{5}}\), we can rationalize the denominator. Rationalizing involves eliminating the square root from the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(1 - \sqrt{5}\) is \(1 + \sqrt{5}\). So, let's proceed with the multiplication:

\[
\frac{\sqrt{8}}{1 - \sqrt{5}} \times \frac{1 + \sqrt{5}}{1 + \sqrt{5}} = \frac{\sqrt{8}(1 + \sqrt{5})}{(1 - \sqrt{5})(1 + \sqrt{5})}
\]

Now, simplify the denominator using the difference of squares formula:

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

Now multiply the numerator:

\[
\sqrt{8}(1 + \sqrt{5}) = \sqrt{8} + \sqrt{8} \times \sqrt{5} = \sqrt{8} + \sqrt{40}
\]

So the expression becomes:

\[
\frac{\sqrt{8} + \sqrt{40}}{-4}
\]

We can simplify the square roots:

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}, \quad \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}
\]

Substituting these into the expression:

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

Factor out the 2 from the numerator:

\[
\frac{2(\sqrt{2} + \sqrt{10})}{-4} = \frac{\sqrt{2} + \sqrt{10}}{-2}
\]

So the simplified expression is:

\[
\boxed{-\frac{\sqrt{2} + \sqrt{10}}{2}}
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you rationalize the denominator when it contains both a square root and a constant?
2. What is the difference of squares formula and how is it used in rationalizing denominators?
3. How do you simplify expressions involving nested square roots?
4. What are some common techniques for simplifying radical expressions?
5. How do you simplify fractions that involve radicals in both the numerator and the denominator?

**Tip:** When simplifying expressions with square roots in the denominator, always consider rationalizing the denominator to remove the square root.

Learn how to simplify the expression \(\frac{\sqrt{8}}{1 - \sqrt{5}}\) using rationalization. This detailed solution covers the process of rationalizing the denominator, applying the difference of squares formula, and simplifying radical expressions. Suitable for students in Grades 10-12.

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