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The question asks for the simplified form of \(\frac{4\sqrt{5}}{3 - \sqrt{5}}\).

To simplify this, we can use the method of multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of \(3 - \sqrt{5}\) is \(3 + \sqrt{5}\).

So, we multiply both the numerator and denominator by \(3 + \sqrt{5}\):

\[
\frac{4\sqrt{5}}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{4\sqrt{5}(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}
\]

Now, let's simplify both the numerator and denominator:

1. **Numerator:**
\[
4\sqrt{5}(3 + \sqrt{5}) = 4\sqrt{5} \cdot 3 + 4\sqrt{5} \cdot \sqrt{5} = 12\sqrt{5} + 4 \cdot 5 = 12\sqrt{5} + 20
\]

2. **Denominator:**
Using the difference of squares formula: 
\[
(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4
\]

Now the fraction becomes:
\[
\frac{12\sqrt{5} + 20}{4}
\]

We can simplify this by dividing both terms in the numerator by 4:
\[
\frac{12\sqrt{5}}{4} + \frac{20}{4} = 3\sqrt{5} + 5
\]

So, the simplified form of \(\frac{4\sqrt{5}}{3 - \sqrt{5}}\) is \(3\sqrt{5} + 5\), which corresponds to option **e** in the choices.

Do you need further details or have any questions?

Here are 5 related questions to consider:
1. What is the importance of multiplying by the conjugate when simplifying expressions with square roots?
2. How would the process change if the denominator were \(3 + \sqrt{5}\) instead of \(3 - \sqrt{5}\)?
3. Can you apply the same method to simplify \(\frac{7\sqrt{3}}{2 - \sqrt{3}}\)?
4. What is the difference between rationalizing the denominator and simplifying radicals?
5. How can you check if your final simplified form is correct?

**Tip:** Always remember to simplify both the numerator and denominator separately before combining terms!

Bentuk sederhana dari \(\frac{4\sqrt{5}}{3 - \sqrt{5}}\) adalah....

Learn how to simplify the expression \(\frac{4\sqrt{5}}{3 - \sqrt{5}}\) using conjugates and algebraic methods. This step-by-step guide covers important algebraic techniques such as the difference of squares and simplifying radical expressions.

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