To simplify the expression $\frac{10}{3\sqrt{5} - 1}$, we rationalize the denominator. Here's the step-by-step process:

Step 1: Multiply numerator and denominator by the conjugate

The conjugate of $3\sqrt{5} - 1$ is $3\sqrt{5} + 1$. Multiplying by this conjugate:

$\frac{10}{3\sqrt{5} - 1} \cdot \frac{3\sqrt{5} + 1}{3\sqrt{5} + 1} = \frac{10(3\sqrt{5} + 1)}{(3\sqrt{5} - 1)(3\sqrt{5} + 1)}$

Step 2: Simplify the denominator

The denominator is in the form of $a^2 - b^2$, which simplifies to:

$(3\sqrt{5})^2 - 1^2 = 45 - 1 = 44$

Step 3: Simplify the numerator

The numerator becomes:

$10(3\sqrt{5} + 1) = 30\sqrt{5} + 10$

Step 4: Combine terms

The entire expression simplifies to:

$\frac{30\sqrt{5} + 10}{44}$

Step 5: Factorize the numerator and denominator

Factoring out $2$:

$\frac{30\sqrt{5} + 10}{44} = \frac{2(15\sqrt{5} + 5)}{44} = \frac{15\sqrt{5} + 5}{22}$

Final Answer:

$\frac{15\sqrt{5} + 5}{22}$

Let me know if you'd like any part explained in more detail!

---

Related Questions:

1. How do you find the conjugate of a binomial expression with a square root?
2. What are some tips for rationalizing denominators involving radicals?
3. Can you explain how to simplify $\frac{1}{\sqrt{5}}$?
4. What is the geometric significance of rationalizing the denominator?
5. How does the difference of squares formula work in simplifying expressions?

Tip:

When rationalizing denominators, always check if the numerator and denominator share common factors that can simplify the final expression further.