Simplifying Surds

Punch \sqrt{2} into a calculator and it spits out 1.41421356\ldots — and it never stops, and it never settles into a repeating pattern. No fraction on Earth equals it exactly. A number like this, a root that refuses to tidy up into a whole number or a fraction, is called a surd. It is irrational: its decimal marches on forever.

So what do mathematicians do with a number they can never write down in full? They refuse to round it. They keep it in exact form — they just write \sqrt{2} and leave it standing. That symbol is the answer, perfect and complete, where 1.41 is only a smudge of it.

But exact form has a tidy version and a messy version. The whole job of this page is turning the messy one into the tidy one. The trick is a single rule about roots of products:

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

Read it right to left and it becomes a tool: if the number under the root hides a perfect square as a factor, you can pull that square out from under the root as an ordinary whole number. 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 the coefficient. And 2\sqrt{3} is exactly the same number as \sqrt{12} — check on a calculator, both are 3.4641\ldots — only now the surd is as simple as it can possibly get.

A surd is a length you can actually draw

Surds are not calculator gremlins — they are ordinary lengths that turn up the moment you draw a slanted line. Take a square exactly 1 unit on each side and draw its diagonal. By Pythagoras, that diagonal is \sqrt{1^2 + 1^2} = \sqrt{2}. It is a real, finite, drawable line — you can measure it with a ruler — yet its exact length is the endless decimal 1.41421356\ldots. That is a surd: perfectly definite, never a fraction.

The recipe

The one word doing the heavy lifting is biggest. Any square factor makes some progress, but only the largest one finishes the job in a single step.

Three worked examples

Example 1 — \sqrt{50}. Scan for square factors: 50 = 25 \times 2, and 25 is a perfect square. Split and pull it out:

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

Example 2 — \sqrt{72}. Here you must hunt for the largest square. It is tempting to grab 72 = 4 \times 18, but that leaves \sqrt{18}, which still has a square hiding in it — so you aren't finished. The biggest square factor of 72 is 36:

\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}

Miss the biggest square and you can still get there — just in two goes: \sqrt{72} = 2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}. Same answer, more work.

Example 3 — a perfect square. Sometimes the whole number under the root is a perfect square, and the surd vanishes completely:

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

If you can't spot the biggest square by eye, break the number into prime factors and pair them up: 72 = 2^3 \times 3^2 = (2 \times 3)^2 \times 2, and each pair of matching primes escapes the root as a single factor.

The splitting rule \sqrt{ab} = \sqrt{a}\,\sqrt{b} works for multiplication under the root — and only multiplication. The single most common surd mistake in every exam is to imagine it works over a plus sign too:

\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}

Don't take my word for it — test it with numbers that are easy to check:

\sqrt{9 + 16} = \sqrt{25} = 5, \quad\text{but}\quad \sqrt{9} + \sqrt{16} = 3 + 4 = 7.

5 \ne 7, so the "rule" is simply false. You may split a root across a times, never across a plus or a minus.

Because 1.41 is a lie by a whisker — it is close, but wrong, and every time you round you throw away a sliver of accuracy that can pile up across a long calculation. \sqrt{2} is right forever. Watch what exactness buys you: \sqrt{2} \times \sqrt{2} = 2, an exact whole number, on the nose. But 1.41 \times 1.41 = 1.9881 — not 2, just near it. Exact surd form is how mathematics carries a number all the way to the end of a problem without ever fudging it.

The ancient Greeks believed, almost as an article of faith, that every length in the universe was a ratio of whole numbers — a fraction. Then someone in the school of Pythagoras proved that the diagonal of a simple unit square, \sqrt{2}, cannot be written as any fraction at all. No \tfrac{p}{q} will ever do it. Legend says the discovery was so disturbing to their whole worldview that they tried to keep it secret — one version has the man who leaked it drowned at sea. A humble square root cracked the foundations of Greek mathematics, and the word for such numbers, irrational, still carries a faint echo of that scandal.

See it explained