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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.

Learn how to simplify the square root expression \( \frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{10 - 2\sqrt{5}}} \) with detailed step-by-step solutions. This problem involves square roots and rationalizing denominators. Suitable for high school students.

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