Simplifying Surds

A surd is a root that can't be written exactly as a whole number or a fraction — like \sqrt{2}, an irrational number whose decimal runs on forever without repeating. We leave it as \sqrt{2} precisely because no fraction is ever exact.

The key rule for working with roots is that a root of a product splits apart:

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

To simplify a surd, split off the largest square factor hiding inside it. Since 12 = 4 \times 3 and 4 is a perfect square:

\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\,\sqrt{3} = 2\sqrt{3}

The 2 out front is called the coefficient, and 2\sqrt{3} is the same number as \sqrt{12} — just written in its simplest form.

The recipe

A root of a product splits: \sqrt{ab} = \sqrt{a}\,\sqrt{b}.
Pull out the biggest perfect-square factor (4, 9, 16, 25, \ldots).
The square root of a perfect square is whole: \sqrt{a^2} = a.
A surd is fully simplified when the number left under the root has no square factors remaining.

For example, hunt for the largest square factor of 50:

\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25}\,\sqrt{2} = 5\sqrt{2}

And when the whole number under the root is a perfect square, the surd disappears entirely:

\sqrt{81} = \sqrt{9^2} = 9