This square root calculator finds the square root (√x) of any number — and when the result is not a whole number, it also simplifies the radical, so √50 becomes 5√2 alongside its decimal value.



√x (decimal)–
Simplified Radical–
x² Check (root × root)–

What Is a Square Root?

The square root of a number is the value that, multiplied by itself, gives the number back: √144 = 12 because 12 × 12 = 144. It is the inverse of squaring, and it is how lengths come back out of areas — the side of a square with area A is √A — which makes it one of the most-used operations in geometry, physics and statistics. This calculator is part of our Math Calculators collection; the most famous root of all lives in the Pythagorean Theorem Calculator (the diagonal is a square root).

Square Root Examples (Verified)

  • √144 = 12 — a perfect square; 12² = 144.
  • √50 ≈ 7.071 — not whole, but simplifiable: 50 = 25 × 2, so √50 = 5√2.
  • √2 ≈ 1.414 — the diagonal of a 1×1 square; irrational, never terminates or repeats.
  • √0.25 = 0.5 — roots of fractions can be tidy too.

Simplifying Radicals (the Textbook Skill)

  • Find the largest perfect-square factor of the number: 72 = 36 × 2.
  • Take its root out front: √72 = 6√2.
  • If no square factor exists (√7, √13), the radical is already simplest.
  • The calculator above performs this factoring automatically for whole numbers up to large values.

For the forward operation (x² and other powers) see our Exponent Calculator — note that √x is the same as x^(1/2), which is how computers compute it.

Square Roots in the Wild

  • Geometry: diagonals, circle radii from areas, the distance formula.
  • Statistics: standard deviation is a square root (of variance) — see our Standard Deviation Calculator.
  • Physics: pendulum periods, escape velocity, RMS values in electronics.
  • Finance: volatility scales with the square root of time (√252 trading days annualizes daily volatility).

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Frequently Asked Questions

What is the square root of 144?

12, because 12 × 12 = 144.

How do I simplify a square root?

Factor the number, pull out any perfect-square factor as its root: √72 = √(36×2) = 6√2. This calculator does it automatically.

What is the square root of a negative number?

An imaginary number: √−9 = 3i, where i = √−1. The calculator shows the imaginary result for negative inputs.

Is the square root of 2 a rational number?

No — √2 is irrational (about 1.41421356…), one of the earliest numbers proven irrational, by the Pythagoreans.

How do I calculate a square root by hand?

The classic method is Newton’s iteration: guess g, then average g with x/g, repeat. Two or three rounds usually give several correct digits.

Is this square root calculator free?

Yes — free, instant, no sign-up.