Surds and Rationalising the Denominator

What Is a Surd?

A surd is a root that cannot be simplified to a rational number: $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$...

$\sqrt{4} = 2$ is NOT a surd (it simplifies to a rational number).

Simplifying Surds

Find the largest square factor:

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

Rules for Surds

• $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
• $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
• $\sqrt{a} \times \sqrt{a} = a$
• $(\sqrt{a})^2 = a$

Adding and Subtracting Surds

Combine like surds only:

$3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

$3\sqrt{2} + 5\sqrt{3}$ — cannot be simplified (different surds).

Rationalising the Denominator

Remove the surd from the denominator.

Simple case: Multiply top and bottom by the surd:

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

With two terms (conjugate): Multiply by the conjugate:

$\frac{1}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$