How to Rationalise Surds

Rationalising surds is about eliminating square roots from denominators because mathematically, we prefer not to have them there. The good news? There are simple techniques that work every time.

For basic surds like $\frac{5}{\sqrt{2}}$, just multiply both top and bottom by the same square root. So you'd multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ to get $\frac{5\sqrt{2}}{2}$. The denominator becomes a regular number because $\sqrt{2} \times \sqrt{2} = 2$.

When you've got more complex denominators like $\frac{3}{4\sqrt{12}}$, the same rule applies. Multiply top and bottom by $\sqrt{12}$ to get $\frac{3\sqrt{12}}{48}$. Remember, having surds in the numerator (top) is perfectly fine - it's only the denominator we're worried about.

Key Point: Always multiply both numerator and denominator by the same value to keep the fraction equivalent.

For tricky expressions like $\frac{1}{(\sqrt{5}+2)}$, you need to use the conjugate method. Multiply by $\frac{(\sqrt{5}-2)}{(\sqrt{5}-2)}$ - notice how we flip the sign. This creates a difference of squares: $(\sqrt{5}+2)(\sqrt{5}-2) = 5-4 = 1$, giving you $\sqrt{5}-2$ as your final answer.