Surds

#### Tier: #Higher

Description

A surd is an irrational root that cannot be simplified to a whole number or fraction. $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ are surds; $\sqrt{4} = 2$ is not.

Simplifying surds: look for square factors. $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$

Multiplying surds: $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$

Rationalising the denominator removes a surd from the bottom of a fraction. For a denominator of the form $a + \sqrt{b}$, multiply top and bottom by $a - \sqrt{b}$ — this uses the Difference of two squares pattern: $$(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7$$

Common error: students write $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$, which is wrong. Only multiplication works that way.

Links

Difference of two squares Algebraic fractions