√(3+√5)/√(3-√5)+√(3-√5)/√(3+√5)

The given expression is:

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

We will simplify this step by step.

### Step 1: Combine the terms into a single fraction
Using the common denominator \(\sqrt{3 - \sqrt{5}} \cdot \sqrt{3 + \sqrt{5}}\), we combine the terms into one fraction:

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

### Step 2: Simplify the denominator
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we simplify the denominator:

\[
(3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4
\]

Thus, the denominator simplifies to \( \sqrt{4} = 2 \).

### Step 3: Simplify the numerator
Now, we simplify the squares in the numerator:

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

So the numerator becomes:

\[
(3 + \sqrt{5}) + (3 - \sqrt{5}) = 3 + \sqrt{5} + 3 - \sqrt{5} = 6
\]

### Step 4: Final expression
Thus, the entire expression becomes:

\[
\frac{6}{2} = 3
\]

### Conclusion:
The value of the given expression is \( \boxed{3} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you handle more complicated radicals in both the numerator and denominator?
2. What happens if both terms under the square roots are negative?
3. How do we rationalize expressions with multiple square roots?
4. Can you apply a similar approach to cubic roots instead of square roots?
5. What is the significance of the identity \((a+b)(a-b) = a^2 - b^2\) in simplifying radicals?

**Tip:** When simplifying radicals, always check if you can apply conjugate pairs or special identities to reduce the complexity.

Learn how to simplify the radical expression √(3+√5)/√(3-√5) + √(3-√5)/√(3+√5) using conjugates and the difference of squares. This detailed solution covers rationalizing denominators and simplifying square roots, suitable for students in Grades 10-12.

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