Radical Simplifier: Simplify Any Square Root

How to use the square root simplifier

1. Enter a non-negative whole number in the input field.
2. The tool automatically factors the number and pulls out every perfect square.
3. Read the simplified radical, the outside coefficient and the remaining radicand below.

How the math works

To simplify a square root, factor the number completely and group identical primes into pairs. Each pair moves outside the radical as a single factor; unpaired primes stay inside.

Take 72 as an example. Its prime factorization is 2 x 2 x 2 x 3 x 3. Group them: (2 x 2) x (3 x 3) x 2. The paired 2 becomes one 2 outside, the paired 3 becomes one 3 outside, and the leftover 2 stays under the radical. Multiply the outside numbers: 2 x 3 = 6. The result is 6√2.

The general formula is √n = a√b where n = a² x b and b has no perfect-square factors greater than 1.

Common square roots simplified

This table shows frequently encountered values you can use to verify homework or check your work quickly.

Why keep the radical form

Converting a square root to a decimal always introduces rounding error. In algebra, geometry and physics, exact answers matter. Writing 6√2 instead of 8.48528... preserves full precision through subsequent calculations.

When you add or subtract radicals, only terms with the same radicand combine. For instance, 3√2 + 5√2 = 8√2. If you had converted to decimals first, you would lose that clean relationship. A radical simplifier makes these like terms visible so you can combine them correctly.

Edge cases and common mistakes

Zero and one: √0 = 0 and √1 = 1. These are already fully simplified.

Prime numbers: A prime like 17 has no perfect-square factors other than 1, so √17 cannot be simplified further. The coefficient is 1 and the radicand remains 17.

Negative inputs: Real square roots of negative numbers do not exist. This tool accepts only non-negative integers. If you need complex numbers, you would write i√|n|, but that is outside the scope here.

Decimals and fractions: Enter only whole numbers. To handle a fraction like 3/4 under a radical, simplify the numerator and denominator separately, then rationalize if needed. For converting decimals to fractions first, try the decimal to fraction tool.

Algorithm behind this tool

The code uses trial division up to √n to find prime factors. For each prime factor found, it counts how many times it divides the number. Every two occurrences contribute one factor to the outside coefficient; any remainder stays inside the radical.

After dividing out all primes up to √n, if the remaining value is greater than 1, it is itself a prime appearing exactly once, so it stays under the radical. This approach handles numbers up to trillions efficiently because the loop only runs to the square root of the input.

Worked examples

Each result below was produced by the same code the tool runs, so you can check your own numbers against them.