simplify leaving no radicals in the denominator. 5 divided by 3 plus the square root of 5

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.

Learn how to simplify the expression \(\frac{5}{3 + \sqrt{5}}\) step-by-step. Rationalize the denominator using the conjugate of \(3 + \sqrt{5}\). Understand the difference of squares formula and its application in simplifying radicals in denominators. Suitable for high school students and those interested in advanced algebra concepts.

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