Surds¶

A surd is an irrational number expressed as a root. A simplified surd should have no square factors.

Rules¶

$\sqrt{ab} = \sqrt{a}\sqrt{b}$
$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Simplifying¶

To simplify a surd, is to extract any rational numbers from it - in practise, removing all square factors.

$\sqrt{8} = \sqrt{4}\sqrt{2} = 2\sqrt{2}$

Multiplying¶

To multiply surds, using the following tip will help: Multiply the non-surd first, and the surd second.

$2\sqrt{3} * 2\sqrt{5} = 4\sqrt{15}$

Adding¶

To add surds, the $a + a + a = 3a$ rule needs to be acknowledged.

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

$\begin{align} \sqrt{2} + \sqrt{8} & = \sqrt{2} + 2\sqrt{2} \\ & = 3\sqrt{2} \end{align}$

Brackets¶

Using brackets and surds is really nothing special.

$(\sqrt{2} + 1)(\sqrt{2} - 1)$

$\begin{array}{c|lc} & \sqrt{2} & +1 \\ \hline \sqrt{2} & 2 & +\sqrt{2} \\ -1 & -\sqrt{2} & -1 \end{array}$

$= 1$

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