To simplify the expression $\frac{5}{3 + \sqrt{5}}$ and remove the radical from the denominator, you should rationalize the denominator. Here’s how to do it:

1. Expression: $\frac{5}{3 + \sqrt{5}}$
2. Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $3 + \sqrt{5}$ is $3 - \sqrt{5}$.

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

3. Simplify the numerator:

$5(3 - \sqrt{5}) = 15 - 5\sqrt{5}$

4. Simplify the denominator using the difference of squares formula:

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

5. Combine the results:

$\frac{15 - 5\sqrt{5}}{4}$

So, the simplified expression is:

$\frac{15 - 5\sqrt{5}}{4}$

Would you like to go over the details, or do you have any questions?

Here are 8 related questions you might find interesting:

1. How do you rationalize a denominator with two different radicals?
2. What is the conjugate of a binomial?
3. How does the difference of squares help in rationalizing denominators?
4. Can you simplify the expression $\frac{7}{2 - \sqrt{3}}$?
5. What happens if the denominator has a cube root instead of a square root?
6. How can you rationalize a denominator that is a binomial with a complex number?
7. What is the simplified form of $\frac{6}{4 + \sqrt{2}}$?
8. Why is it necessary to remove radicals from the denominator in certain cases?

Tip: When rationalizing denominators, always use the conjugate of the denominator to eliminate the radical.