Simplify Square Root Calculator

What simplest radical form means

A square root is in simplest radical form when the number under the radical has no perfect square factor other than 1. √72 is not: 72 = 36 × 2, and the 36 can come out as its root, 6. What stays is 6√2 — the same number, written exactly and with the smallest possible radicand.

Where the formula comes from

For non-negative numbers the root of a product is the product of the roots: √(x × y) = √x × √y. Split N into its largest perfect square factor a² and the rest b, and the first root is a whole number, because √(a²) = a for a ≥ 0:

The rule holds only for x, y ≥ 0 — that is why this page simplifies square roots of non-negative numbers and nothing else.

How to simplify a square root in three steps

1. Factor the number into primes: 72 = 2 × 2 × 2 × 3 × 3.
2. Group equal primes into pairs: (2 × 2) × (3 × 3) × 2. Every pair is a perfect square.
3. Take one prime of every pair out of the root and multiply them; leave the unpaired primes under it: 2 × 3 × √2 = 6√2.

Worked example: √72

The same three steps written out in full, with the check that catches a wrong answer:

A shortcut that works for small numbers: walk the list of perfect squares below from the bottom up — largest first — and stop at the first one that divides the number. For 72 that is 36, and 72 ÷ 36 = 2 gives √72 = 6√2 directly. Taking a smaller square instead — 4 or 9 — gives 2√18 or 3√8, which are correct but not yet simplest: both radicands still hide a square factor.

Perfect squares from 4 to 400

Common square roots in simplest radical form

Where simplified radicals are used

• School algebra — answers are expected in exact form: 6√2, not 8.485
• Pythagorean theorem — the diagonal of a square with side 3 is √(3² + 3²) = √18 = 3√2
• Equilateral triangles — the height of one with side 4 is √(4² − 2²) = √12 = 2√3
• Quadratic equations — x² − 6x + 1 = 0 has discriminant 32, √32 = 4√2, so x = 3 ± 2√2
• Adding radicals — only like radicals add up: √8 + √18 = 2√2 + 3√2 = 5√2
• Distance between points — from (0, 0) to (2, 4) it is √(2² + 4²) = √20 = 2√5

Limits and caveats

• Non-negative numbers only. √−8 is not a real number; in the complex numbers it is 2i√2, and this page does not compute it.
• Whole numbers only. Roots of fractions and decimals, such as √(1/2) = √2 ÷ 2, need rationalizing the denominator — a different procedure.
• Square roots only. A cube root comes out by triples of equal primes, not pairs: ∛16 = 2∛2.
• The decimal is a check, not the answer. √2 is irrational, so every decimal of 6√2 is rounded; the simplified radical is the exact value.
• Range. The form accepts whole numbers from 1 to 2,147,483,643.