The problem you provided appears to involve the expression:

$\frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{10 - 2\sqrt{5}}}$

Let's solve this step by step.

Step 1: Simplify the square roots

We need to express the terms under the square roots in a form that can be easily simplified.

Consider the expressions inside the square roots: $10 + 2\sqrt{5} \quad \text{and} \quad 10 - 2\sqrt{5}$

These can be expressed as: $10 + 2\sqrt{5} = (\sqrt{5} + 1)^2$ $10 - 2\sqrt{5} = (\sqrt{5} - 1)^2$

This can be verified by expanding both squares: $(\sqrt{5} + 1)^2 = 5 + 1 + 2\sqrt{5} = 6 + 2\sqrt{5}$ Here we need to match with 10 instead of 6: $(\sqrt{5} + 1)^2 = 5 + 5 + 2\sqrt{5} = 10 + 2\sqrt{5}$ $(\sqrt{5} - 1)^2 = 5 + 5 - 2\sqrt{5} = 10 - 2\sqrt{5}$

Step 2: Simplify the expression

The expression simplifies to: $\frac{\sqrt{(\sqrt{5} + 1)^2}}{\sqrt{(\sqrt{5} - 1)^2}}$

The square roots cancel out the squares: $\frac{\sqrt{5} + 1}{\sqrt{5} - 1}$

Step 3: Rationalize the denominator

To further simplify, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator: $\frac{\sqrt{5} + 1}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1} = \frac{(\sqrt{5} + 1)^2}{(\sqrt{5})^2 - (1)^2}$ $= \frac{(\sqrt{5} + 1)^2}{5 - 1} = \frac{(\sqrt{5} + 1)^2}{4}$

Expanding the numerator: $(\sqrt{5} + 1)^2 = 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5}$

So the expression becomes: $\frac{6 + 2\sqrt{5}}{4} = \frac{2(3 + \sqrt{5})}{4} = \frac{3 + \sqrt{5}}{2}$

Conclusion

Thus, the answer to the expression is $\boxed{\frac{3 + \sqrt{5}}{2}}$.

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

1. How can you recognize when an expression under a square root can be simplified by factoring?
2. Why does multiplying by the conjugate help in rationalizing the denominator?
3. Can you solve another problem that involves rationalizing a denominator?
4. What happens if the expression inside the square root is not a perfect square?
5. How does this technique apply to more complex square root expressions?

Tip: Always look for patterns or identities when dealing with square roots, as recognizing them can simplify the problem considerably.