Square Root Calculator

The Square Root Calculator finds the principal (positive) and negative square roots of any non-negative number, checks whether it's a perfect square, and — for whole numbers — simplifies the result to its radical form (like √72 = 6√2).

It's built for students checking homework, teachers preparing examples, and anyone who needs a square root without reaching for a separate radical-simplification step.

How to use it

1. Enter a number that is zero or positive. Square roots of negative numbers aren't real, so those are rejected.
2. Read the Principal Square Root and Negative Square Root fields for the two roots.
3. Check Perfect Square? to see whether the number is a perfect square, and Simplified Radical Form for the radical written in simplest terms.

Results update automatically as you type. For example, entering 72 returns a principal root of 8.48528, a negative root of -8.48528, and a simplified radical form of 6√2.

What is a principal square root?

Every positive number has two square roots — one positive, one negative — since squaring either gives the same result (e.g. 4² = 16 and (-4)² = 16). The principal square root is the positive one, and it's what the √ symbol refers to by convention. The negative root is included here for completeness, since some equations (like x² = n) need both.

What is a perfect square?

A perfect square is a number that's the square of an integer — 1, 4, 9, 16, 25, and so on. Its square root is a whole number with no remaining radical. Any other non-negative number has an irrational square root, which can only be approximated as a decimal or written in simplified radical form.

What does "simplified radical form" mean?

For a whole number that isn't a perfect square, the square root can often still be simplified by pulling the largest perfect-square factor out from under the radical. For example, 72 = 36 × 2, and 36 is a perfect square, so √72 = √36 × √2 = 6√2 — a shorter, exact form instead of an infinite decimal. Numbers with no perfect-square factor greater than 1 (like 7 or 11) are already in simplest form.

Why can't I enter a negative number?

The square root of a negative number isn't a real number — it's an imaginary number (a multiple of i, where i = √-1), which is outside the scope of this calculator. For imaginary and complex roots, use the Quadratic Formula Solver, which handles complex conjugate roots directly.

To see the prime factors this calculator's simplification step is built on, try the Prime Factorization Calculator.

Privacy

This calculator runs entirely in your browser. Your number is never uploaded, logged, or stored — the computation happens locally on your device.