Quick answer: A square root asks which number multiplied by itself gives the target: the square root of 25 is 5. Perfect squares like 1, 4, 9 and 16 have whole-number roots. Simplify radicals by factoring squares out, so root 50 becomes 5 root 2. Cube roots work the same way for three dimensions: the cube root of 27 is 3.
Last reviewed: September 2026
Key facts
- √144 = 12; √2 ≈ 1.41421356 — the first irrational number discovered.
- The square root asks: “which number, times itself, gives this?”
- Every positive number has two square roots (+ and −); calculators give the positive one.
- A square room of 25 m² has sides of exactly 5 m.
The square root undoes squaring: it finds the number which, multiplied by itself, gives your input. This free square root calculator returns √x instantly for any non-negative number, with a perfect-squares table below for mental maths.
Enter a number and read the root. It is the tool for geometry (side from area), statistics (standard deviation) and the quadratic formula.
How to use
- Enter a number (zero or positive).
- The square root appears instantly.
- Check perfect squares in the table below for mental-maths practice.
- For other roots (cube, nth), use the Exponent Calculator with a fractional exponent.
Square root definition
| Term | Meaning |
|---|---|
| x | The number under the root (must be ≥ 0 for real results). |
| y | The root — the calculator returns the non-negative one. |
Why no negative inputs? No real number squared gives a negative — √(−1) is the imaginary unit i, beyond this calculator’s scope.
Worked examples
Perfect square
√144: 12 × 12 = 144, so √144 = 12.
Geometry use: a square garden of area 144 m² has sides of 12 m.
Irrational root
√2 ≈ 1.41421356 — it never terminates or repeats. The Pythagoreans found this shocking: the diagonal of a 1×1 square cannot be written as a fraction.
Perfect squares to memorise
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 8 | 64 |
| 2 | 4 | 9 | 81 |
| 3 | 9 | 10 | 100 |
| 4 | 16 | 11 | 121 |
| 5 | 25 | 12 | 144 |
| 6 | 36 | 13 | 169 |
| 7 | 49 | 14 | 196 |
Frequently asked questions
What is the square root of a negative number?
Not a real number — it is imaginary (√(−1) = i). This calculator works with real numbers only.
Why does the calculator give only the positive root?
By convention √x means the principal (non-negative) root. The equation x² = 9 has two solutions, ±3, but √9 = 3.
Is the square root of 2 rational?
No — √2 ≈ 1.41421356… never repeats. Its irrationality was proved in ancient Greece.
How do I find a cube root?
Use the Exponent Calculator: cube root of x = x1/3.
Where are square roots actually used?
Geometry (side from area), the quadratic formula, standard deviation in statistics, and the Pythagorean theorem: c = √(a² + b²).
Sources
- Khan Academy — Square and cube roots
- Maths is Fun — Roots with worked examples
Key takeaways
- √x is the non-negative number that squares to x.
- √2 ≈ 1.41421… is irrational — it never repeats.
- Negative inputs have no real root; fractional exponents give cube/nth roots.
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