The expression in the image is:

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

To simplify this, let's proceed with the following steps:

Step 1: Rationalize the Denominator

Multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}$:

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

Step 2: Simplify the Denominator

The denominator is now in the form of a difference of squares: $\left(\sqrt{3 + \sqrt{5}}\right)^2 - \left(\sqrt{3 - \sqrt{5}}\right)^2$ Calculating each term separately:

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

Thus, the denominator simplifies to: $(3 + \sqrt{5}) - (3 - \sqrt{5}) = 2\sqrt{5}$

Step 3: Simplify the Numerator

The numerator is: $\left(\sqrt{3 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}\right)^2$

Expanding this: $(\sqrt{3 + \sqrt{5}})^2 + 2\sqrt{(3 + \sqrt{5})(3 - \sqrt{5})} + (\sqrt{3 - \sqrt{5}})^2$

We already know that:

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

Now, let’s calculate $2\sqrt{(3 + \sqrt{5})(3 - \sqrt{5})}$.

Using the difference of squares: $(3 + \sqrt{5})(3 - \sqrt{5}) = 9 - 5 = 4$

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

Thus, the numerator simplifies to: $(3 + \sqrt{5}) + (3 - \sqrt{5}) + 4 = 6 + 4 = 10$

Final Answer

The expression simplifies to: $\frac{10}{2\sqrt{5}} = \frac{10}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{10} = \sqrt{5}$

So, the simplified form is: $\sqrt{5}$

Would you like further details on any step? Here are some related questions:

1. How does rationalizing the denominator work in simplifying expressions?
2. Why is it useful to multiply by the conjugate in such problems?
3. What are common techniques for simplifying expressions with nested radicals?
4. How does the difference of squares formula help in these calculations?
5. Can we apply a similar method to other radical expressions?

Tip: For complex expressions with radicals, look for opportunities to use conjugates to simplify the problem.