Without using a Calculator or a Four Figure table, simplify the surd and leave your answer in rationalised denominator.

We have $\sqrt{100} = 10$ and $\sqrt{169} = 13$.
Substitute these values into the given expression:

$$\frac{4\sqrt{100}}{3\sqrt{169} - \sqrt{39}} = \frac{4 \times 10}{3 \times 13 - \sqrt{39}}$$

To rationalize the denominator $39 - \sqrt{39}$, multiply both the numerator and the denominator by its conjugate, which is $39 + \sqrt{39}$.

$$\frac{40}{39 - \sqrt{39}} \times \frac{39 + \sqrt{39}}{39 + \sqrt{39}}$$

For the numerator: $40(39 + \sqrt{39})$
For the denominator, use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:

$$(39 - \sqrt{39})(39 + \sqrt{39}) = 39^2 - (\sqrt{39})^2$$

$$39^2 = 1521$$ $$(\sqrt{39})^2 = 39$$

So the denominator becomes:

$$1521 - 39 = 1482$$

Both 40 and 1482 are divisible by 2.

$$40 \div 2 = 20$$ $$1482 \div 2 = 741$$

The simplified expression is:

$$\frac{20(39 + \sqrt{39})}{741}$$