Rationalise The Denominator

Rationalising the denominator means rewriting a fraction so that the denominator does not contain any surds (irrational numbers). This process makes the fraction easier to work with, particularly in further algebraic operations.

Rationalising with a Single Surd

If the denominator contains a single surd, multiply both the numerator and the denominator by that surd.

$$\frac{1}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}$$

Rationalising with a Binomial Surd (Difference of Squares)

When the denominator is in the form $a+\sqrt{b}$ or $a-\sqrt{b}$​, multiply the numerator and denominator by the conjugate. The conjugate is the same binomial, but with the opposite sign between the terms.

$$\frac{1}{2+\sqrt{5}}\times \frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{2-\sqrt{5}}{(2+\sqrt{5})(2-\sqrt{5})}$$

Use the difference of squares formula:

$$(2+\sqrt{5})(2-\sqrt{5})=2^2-(\sqrt{5}{)}^2=4-5=-1$$

So:

$$\frac{2-\sqrt{5}}{2+\sqrt{5}}=\frac{2-\sqrt{5}}{-1}=-(2-\sqrt{5})=-2+\sqrt{5}$$