Square Root Calculator — find √x

How it works

The square root of x is the number that squares to give x — the reverse of squaring. Written √x (or x^(1/2)), it answers "what times itself equals this?":

√x = r where r × r = x

Every positive number technically has two square roots, one positive and one negative (since (−3)² = 9 too), but "the" square root normally means the principal, non-negative one, which is what this calculator returns. Perfect squares — 1, 4, 9, 16, 25, … — have whole-number roots; everything else has an irrational root that never terminates or repeats.

Worked example

Find √144. Ask which number times itself makes 144 — that's 12, since 12 × 12 = 144, so it's a perfect square. Most numbers aren't so tidy: √2 is 1.41421356…, an irrational number, and √50 ≈ 7.0711. You can always check a root by squaring it back: 7.0711² ≈ 50.

Frequently asked questions

How do I find a square root?

For a perfect square, find the whole number that squares to it (√81 = 9 since 9² = 81). For other numbers the root is irrational, so you estimate or compute it — √20 is between √16 = 4 and √25 = 5, and works out to about 4.472. This calculator computes it to several decimal places.

Why can't you take the square root of a negative number?

Because any real number squared is positive or zero — there's no real value whose square is negative. So √(−9) has no real answer. Mathematics extends to "imaginary" numbers (√(−1) = i) to handle these, but those aren't real numbers; this calculator works with real inputs and leaves negatives blank.

Does a number have one square root or two?

Two: a positive and a negative, since both (+3)² and (−3)² equal 9. The **principal square root** is the non-negative one, which is what √ usually denotes and what this tool gives. When solving an equation like x² = 9, remember to include both, x = ±3.

What's a perfect square?

A number that's the square of a whole number: 1, 4, 9, 16, 25, 36, … Their square roots are exact integers. Any other positive number has an irrational square root — a decimal that goes on forever without a repeating pattern, like √3 = 1.7320508…

How is a square root related to exponents?

A square root is the ½ power: √x = x^(1/2). That's why it undoes squaring (x²) — raising to ½ and then to 2 gets you back to x. More generally, the nth root is the 1/n power.

How can I estimate a square root by hand?

Find the two perfect squares it sits between and interpolate. For √40: it's between √36 = 6 and √49 = 7, and closer to 6, so roughly 6.3 — the exact value is 6.3246. Squaring your guess and adjusting quickly narrows it down.