Surd Calculator

How it works

A surd is the Australian and UK maths-curriculum term for a square root (or other root) that can’t be simplified into a whole number or an exact fraction — its decimal value runs on forever without repeating, because the number underneath is irrational. √72 is a surd. √16 is not, because it simplifies exactly to 4.

“Simplifying” a surd doesn’t get rid of the root sign — a true surd’s exact value can never be written as a finite decimal. What it means is rewriting the number under the root (the radicand) using the smallest possible number left inside, by pulling any perfect-square factor out the front. This is often called simplest radical form, or simplest surd form:

If n = a² × b, then √n = a√b

To simplify properly you need the largest perfect-square factor of the radicand — a smaller one still works but leaves more simplifying to do. This calculator finds it the same way you would by hand under exam conditions: starting from the whole-number part of √n and testing downward, it checks each candidate i (from ⌊√n⌋ down to 1) to see whether i² divides n exactly. The first i² that does is the largest perfect-square factor — a√b then has a = i and b = n ÷ i².

This tool only handles square roots (surds of degree 2), which covers what’s taught in Australian junior and senior maths courses. It doesn’t attempt to simplify cube roots or higher-degree roots, which follow a different rule.

If you type in a number that isn’t a whole number, the calculator rounds it down to simplify (perfect-square factoring only makes sense for whole numbers), but the decimal value shown alongside is always the true square root of exactly what you entered — see Math Is Fun’s guide to simplifying radicals for more worked examples of the same method.

Worked example

Ravi, a Year 10 student in Perth, hits a “simplify √72” question on a practice test and needs to show full working, not just a decimal answer.

He knows the rule: find the largest perfect-square factor of 72, then split it out of the root. Working down from the nearest whole number below √72 (which sits between 8 and 9, since 8² = 64 and 9² = 81), he tests each candidate in turn:

Candidate	Square	Divides 72 evenly?
8	64	No — 72 ÷ 64 = 1.125
7	49	No — 72 ÷ 49 ≈ 1.469
6	36	Yes — 72 ÷ 36 = 2

36 is the first, and therefore largest, perfect-square factor that divides 72 exactly, so Ravi rewrites 72 as 36 × 2 and splits the root:

√72 = √(36 × 2) = √36 × √2 = 6√2

As a check, both 6√2 and √72 work out to the same decimal — approximately 8.4853. But because 8.4853 is only a rounded approximation and 6√2 is the exact value, Ravi writes 6√2 as his final answer to pick up full marks on the test.

	Value
Radicand	72
Largest perfect-square factor	36 (6²)
Simplified surd	6√2
Decimal (≈)	8.4853