The expression given is:

$\frac{\sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}}{\sqrt{3 + \sqrt{5}} - \sqrt{3 - \sqrt{5}}}$

Let's simplify this expression step-by-step.

Step 1: Rationalize the Denominator

To simplify, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}$.

The expression becomes:

$\frac{\left( \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}} \right) \cdot \left( \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}} \right)}{\left( \sqrt{3 + \sqrt{5}} - \sqrt{3 - \sqrt{5}} \right) \cdot \left( \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}} \right)}$

Step 2: Simplify the Denominator Using the Difference of Squares

The denominator becomes a difference of squares:

$\left( \sqrt{3 + \sqrt{5}} \right)^2 - \left( \sqrt{3 - \sqrt{5}} \right)^2$

Calculating each term:

1. $\left( \sqrt{3 + \sqrt{5}} \right)^2 = 3 + \sqrt{5}$
2. $\left( \sqrt{3 - \sqrt{5}} \right)^2 = 3 - \sqrt{5}$

So the denominator simplifies to:

$(3 + \sqrt{5}) - (3 - \sqrt{5}) = 2\sqrt{5}$

Step 3: Expand the Numerator

The numerator, after multiplying out, becomes:

$\left( \sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}} \right)^2$

Expanding this square:

$= \left( \sqrt{3 + \sqrt{5}} \right)^2 + 2 \cdot \sqrt{3 + \sqrt{5}} \cdot \sqrt{3 - \sqrt{5}} + \left( \sqrt{3 - \sqrt{5}} \right)^2$

Calculating each part:

1. $\left( \sqrt{3 + \sqrt{5}} \right)^2 = 3 + \sqrt{5}$
2. $\left( \sqrt{3 - \sqrt{5}} \right)^2 = 3 - \sqrt{5}$
3. $2 \cdot \sqrt{(3 + \sqrt{5})(3 - \sqrt{5})}$

Now, $(3 + \sqrt{5})(3 - \sqrt{5})$ simplifies as follows:

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

So the middle term becomes:

$2 \cdot \sqrt{4} = 2 \cdot 2 = 4$

Adding these together:

$(3 + \sqrt{5}) + (3 - \sqrt{5}) + 4 = 6 + 4 = 10$

Step 4: Put it All Together

The simplified form of the expression is:

$\frac{10}{2\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}$

Final Answer

$\sqrt{5}$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How do you simplify an expression involving square roots and conjugates?
2. What is the purpose of rationalizing the denominator?
3. How does the difference of squares help in simplifying square root expressions?
4. What is a conjugate, and why is it useful in these kinds of problems?
5. How can the properties of radicals be used to simplify complex expressions?

Tip: When simplifying expressions with square roots, look for opportunities to use conjugates and the difference of squares to eliminate radicals in the denominator.