Question: Rationalise the denominator of $\frac{15 + \sqrt{5}}{6\sqrt{5}}$
Give your answer in its simplest form.
1. **State the problem:** Rationalise the denominator of the expression $$\frac{15 + \sqrt{5}}{6\sqrt{5}}$$ and simplify the result.
2. **Recall the rule:** To rationalise a denominator containing a square root, multiply numerator and denominator by the conjugate or an expression that removes the root from the denominator.
3. **Multiply numerator and denominator by $\sqrt{5}$** to eliminate the square root in the denominator:
$$\frac{15 + \sqrt{5}}{6\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{(15 + \sqrt{5})\sqrt{5}}{6 \cancel{\sqrt{5}} \cancel{\sqrt{5}}} = \frac{(15 + \sqrt{5})\sqrt{5}}{6 \times 5}$$
4. **Simplify the denominator:**
$$6 \times 5 = 30$$
5. **Expand the numerator:**
$$ (15 + \sqrt{5})\sqrt{5} = 15\sqrt{5} + \sqrt{5} \times \sqrt{5} = 15\sqrt{5} + 5 $$
6. **Write the expression:**
$$\frac{15\sqrt{5} + 5}{30}$$
7. **Factor numerator if possible:**
$$5(3\sqrt{5} + 1)$$
8. **Simplify the fraction by dividing numerator and denominator by 5:**
$$\frac{\cancel{5}(3\sqrt{5} + 1)}{\cancel{5} \times 6} = \frac{3\sqrt{5} + 1}{6}$$
**Final answer:**
$$\boxed{\frac{3\sqrt{5} + 1}{6}}$$