How do you rationalise the denominator of 8/(4 + √5)?

To rationalise the denominator of \( \frac{8}{4 + \sqrt{5}} \), multiply by the conjugate \( \frac{4 - \sqrt{5}}{4 - \sqrt{5}} \).

Rationalising the denominator involves removing any surds (square roots) from the bottom of a fraction. To do this, we use the conjugate of the denominator. The conjugate of \( 4 + \sqrt{5} \) is \( 4 - \sqrt{5} \). By multiplying the numerator and the denominator by this conjugate, we can eliminate the surd from the denominator.

Here's the step-by-step process:

1. Write the original fraction: \( \frac{8}{4 + \sqrt{5}} \).
2. Multiply both the numerator and the denominator by the conjugate of the denominator: \( \frac{4 - \sqrt{5}}{4 - \sqrt{5}} \).

This gives us:
\[ \frac{8 \cdot (4 - \sqrt{5})}{(4 + \sqrt{5}) \cdot (4 - \sqrt{5})} \]

3. Expand the numerator and the denominator:
\[ 8 \cdot (4 - \sqrt{5}) = 32 - 8\sqrt{5} \]
\[ (4 + \sqrt{5}) \cdot (4 - \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11 \]

4. Combine the results:
\[ \frac{32 - 8\sqrt{5}}{11} \]

So, the rationalised form of \( \frac{8}{4 + \sqrt{5}} \) is \( \frac{32 - 8\sqrt{5}}{11} \).