Exact lengths keep turning up as surds. The diagonal across a one-metre square, the height of an equilateral triangle, the longer side of an A4 sheet against its shorter one — none of these are whole numbers or tidy decimals; they are square roots. To add such lengths, or find an area, without rounding away their exactness, you need to combine surds cleanly. That is what this page is about.

Here is the secret that makes surd arithmetic easy: treat the root like a letter. Once you can simplify a surd, you can add, subtract and multiply surds by the very same moves you already use in algebra. Think of \sqrt{3} as if it were a variable x. Then

Everything on this page is that one idea, pushed a little further: collect like surds the way you collect like terms, and multiply roots with a single tidy rule.

Adding and subtracting: collect like surds

You can only add or subtract like surds — surds with the same number under the root. The root is the shared symbol; you just add the coefficients in front of it.

Unlike surds refuse to merge. \sqrt{2} + \sqrt{3} stays exactly as it is — there is no single surd equal to it, just as 2x + 5y won't collapse into one term.

The clever bit: surds often only look unlike. Simplify each one first and hidden like surds appear. Watch \sqrt{8} and \sqrt{18} — apparent strangers — both turn into multiples of \sqrt{2}:

So the golden rule for a sum of surds is: simplify first, then collect.

Multiplying: roots combine under one sign

To multiply surds, multiply the numbers under the roots together — that is the splitting rule run backwards:

So \sqrt{3}\times\sqrt{12} = \sqrt{36} = 6. When there are coefficients out front, gather them separately — numbers with numbers, roots with roots:

And the neatest fact of all: a surd times itself destroys the root entirely — because that is precisely what a square root means:

Brackets expand exactly as in algebra — multiply each part out, then simplify what you can:

Notice the first term lost its root (a surd times itself) while the second kept it.

Putting it all together

Real problems mix the moves. Take \sqrt{3}\,(\sqrt{12} - \sqrt{3}). Work left to right, one rule at a time, and keep everything exact:

A whole tangle of roots collapsed to the plain number 3 — and it is exactly 3, not "about 3", because nothing was ever rounded. Here is one more, blending simplify-then-collect with a multiplication:

The \sqrt{12} had to be simplified before its hidden \sqrt{3} could join the party. That is the whole game: multiply and split as needed, simplify, then collect like surds.