Operations with Surds

Adding and subtracting surds

You can only add or subtract like surds — surds with the same root. It works exactly like collecting like terms in algebra: the root behaves like a shared symbol, and you just add the numbers in front.

2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}

Unlike surds cannot be combined into a single surd. For example \sqrt{2} + \sqrt{3} stays as it is — there is no simpler surd equal to it.

Sometimes surds only look different. Simplify each one first, and they may turn out to be like surds after all:

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

Multiplying surds

To multiply surds, multiply what is under the roots together:

\sqrt{a}\times\sqrt{b} = \sqrt{ab}

So \sqrt{3}\times\sqrt{12} = \sqrt{36} = 6. A surd times itself just removes the root: \sqrt{a}\times\sqrt{a} = a, for example \sqrt{5}\times\sqrt{5} = 5.

Brackets expand exactly as they do in algebra — multiply each part out, then simplify:

\sqrt{2}\,(\sqrt{2} + \sqrt{6}) = \sqrt{2}\times\sqrt{2} + \sqrt{2}\times\sqrt{6} = 2 + \sqrt{12} = 2 + 2\sqrt{3}