Square Root Calculator
Find the square root of any non-negative number, shown to 10 significant digits, with whole numbers also simplified to radical form (√50 = 5√2).
Last updated: October 2026
Square root
√50 = 5√2 ≈ 7.071067812
Formula below · 2 sources (mathworld.wolfram.com, mathsisfun.com) · Updated Oct 2026
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About this calculator
The square root of n is the non-negative number that gives n when multiplied by itself: √n × √n = n. So √144 = 12, because 12 × 12 = 144. Every positive number also has a negative square root (−12 × −12 = 144); like most calculators, this one returns the positive one, called the principal root.
Exact, simplified and decimal answers
Perfect squares (1, 4, 9, 16, 25, …) have whole-number roots. For other whole numbers the root is irrational, so the calculator shows two forms: the simplified radical and a decimal to 10 significant digits. To simplify, it pulls out the largest square factor: 50 = 25 × 2, so √50 = √25 × √2 = 5√2 ≈ 7.071067812. A number with no square factor, such as √30, is already in simplest form and is shown as a decimal only.
Negative numbers
No real number squared gives a negative result, so the square root of a negative number is not a real number. In complex numbers, √(−9) = 3i. This calculator accepts 0 and positive numbers.
| n | √n exact | √n decimal |
|---|---|---|
| 2 | √2 | 1.414213562 |
| 3 | √3 | 1.732050808 |
| 8 | 2√2 | 2.828427125 |
| 12 | 2√3 | 3.464101615 |
| 18 | 3√2 | 4.242640687 |
| 50 | 5√2 | 7.071067812 |
| 75 | 5√3 | 8.660254038 |
| 200 | 10√2 | 14.14213562 |
How to use
Type the number and read the root. Example 1: 144 gives √144 = 12 (a perfect square). Example 2: 50 gives √50 = 5√2 ≈ 7.071067812. Check: 7.071067812² ≈ 50. Example 3: 72 gives √72 = 6√2 ≈ 8.485281374, because 72 = 36 × 2. Example 4: 2.25 gives √2.25 = 1.5. A square garden of 2.25 m² has 1.5 m sides.
Frequently asked questions
How do you simplify a square root?
Find the largest perfect square that divides the number, take its root outside, and leave the rest under the radical. √72 = √(36 × 2) = 6√2. If no square larger than 1 divides the number, the root is already simplified.
How do I estimate a square root without a calculator?
Find the two perfect squares around the number. √50 lies between √49 = 7 and √64 = 8, and much closer to 7. A quick refinement: average your guess g with n ÷ g. For g = 7, (7 + 50/7) ÷ 2 ≈ 7.0714, already correct to three decimal places.
Why can't I take the square root of a negative number?
Any real number times itself is zero or positive, so no real number squares to −9. Mathematicians define i = √(−1), which makes √(−9) = 3i, a complex number.
Is the square root of 2 exactly 1.414?
No. √2 is irrational, so its decimal never ends or repeats. 1.414 is a rounded value; to 10 significant digits it is 1.414213562.
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Sources & references
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