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Square Root Calculator (Simplified Radical, Negative Numbers)

Square root of any number: decimal, simplified radical, and imaginary roots for negatives.

Inputs

√500
22.36068

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Free embed

  • Two lines of HTML, unlimited views
  • Shows a small CalculateEveryday bar, with a credit link under the calculator

What each Square Root input means

Number (negative allowed)
Any number from −1 billion to 1 billion, decimals included. A negative number gives an imaginary root written with i, and a whole number that is not a perfect square also gets its simplified radical form, such as 6√2.

Square Root formula

√n = the non-negative x with x² = n · simplify: √(a²·b) = a√b · negative: √(−n) = √n · i

a2b=ab−n=in\sqrt{a^2 b} = a\sqrt{b} \qquad \sqrt{-n} = i\sqrt{n}

The square root of a number is the value that, multiplied by itself, gives that number back. Every positive number has two — one positive, one negative — and the radical sign means the positive one, the principal root, which is what the headline shows; the line beneath gives both. When the number is a whole number that is not a perfect square, the page also writes the root in simplified radical form by pulling out the largest perfect-square factor: 500 = 100 × 5, so √500 = √100 × √5 = 10√5, which is the exact answer a maths class usually wants. A negative number has no real square root, because no real number squared is negative; its root is imaginary, written with i where i² = −1: √−49 = √49 × √−1 = 7i.

Square Root worked example

√500: 500 is not a perfect square (22² = 484 and 23² = 529), so the root is a decimal between 22 and 23 — 22.36068. For the exact form, find the largest perfect square that divides 500: that is 100, so √500 = √(100 × 5) = 10√5. The squared check multiplies 22.36068 by itself and gets 500.000000.

More Square Root examples

√−8: no real number squares to −8, so the root is imaginary. √8 = 2√2, so √−8 = 2i√2, and the headline shows 2.828427i (≈), the mark saying the decimal is rounded. The other square root is −2.828427i.

√72: 72 = 36 × 2, so √72 = 6√2, which is 8.485281 — between √64 = 8 and √81 = 9.

√0.25 = 0.5, because 0.5 × 0.5 = 0.25. The square root of a number between 0 and 1 is larger than the number itself.

Where the Square Root calculator can give the wrong answer

  • The headline is the principal (positive) root. If an equation says x² = 25, both 5 and −5 solve it; the 'both square roots' line is there for that case.
  • Simplified radical form is only offered for whole numbers. A decimal like 2.5 is shown as a decimal root (1.581139) — there is no meaningful a√b form for it.
  • Whole numbers are checked for perfect-square factors by trial division, which is instant up to the billion allowed here but is not a route to factoring very large numbers.
  • Floating point arithmetic cannot represent every decimal exactly, so the squared check can read 143.999999 instead of 144. That is display rounding at the sixth decimal, not a wrong root.
  • An imaginary result such as 7i is exact only for a perfect square; √−2 is shown as 1.414214i with an approximation mark, and its exact form is i√2.

Spotted a mistake? Tell us

Square Root FAQ

For a perfect square, find the number that multiplies by itself to give it: 12 × 12 = 144, so √144 = 12. For anything else, bracket it between perfect squares — √50 is between √49 = 7 and √64 = 8, so a little over 7 — then refine: 7.1² = 50.41 is too big, 7.07² = 49.98 is close. The page does the same and shows the bracket.

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