Surds

What a surd is

• A surd is a root that cannot be simplified to a whole number — its decimal goes on forever without repeating (it is irrational), like $\sqrt{2} = 1.41421\ldots$
• We keep surds in root form because they are exact: $\sqrt{2}$ is precise, whereas $1.41$ is only a rounded approximation that loses accuracy.
• Not every root is a surd: $\sqrt{9} = 3$ is a whole number, so it is not a surd.

Simplifying a surd

• The idea is to pull out any perfect-square factor (a factor like $4, 9, 16, 25, \ldots$) hiding under the root.
• Find the largest perfect-square factor of the number.
• Split the root using $\sqrt{a \times b} = \sqrt{a}\,\sqrt{b}$, then take the root of the perfect square:

$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4}\,\sqrt{5} = 2\sqrt{5}$

Adding, subtracting and multiplying surds

• Add or subtract only like surds (the same number under the root), exactly like collecting like terms: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$.
• Multiply using $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.
• A surd times itself removes the root: $\sqrt{a} \times \sqrt{a} = a$ (e.g. $\sqrt{7} \times \sqrt{7} = 7$).

Rationalising the denominator

• Rationalising means removing a surd from the bottom of a fraction (the convention is to leave denominators surd-free).
• Multiply the top and bottom by the surd on the bottom — this is really multiplying by 1, so the value is unchanged, but the bottom becomes a whole number:

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$