# Free online square root calculator

> A square root is the number that, when multiplied by itself, gives the original value — √9 = 3 because 3×3 = 9. This calculator finds the nth root of any real number and shows.

Calculate the square root, cube root, or any nth root of any real number. Shows whether the result is a perfect root, and includes the value squared and cubed for reference.

**Interactive version:** https://onlinecalculator.me/math/square-root/
**Category:** Math

## How to use

1. Enter the value whose root you want.
2. Set the root degree — 2 for square root, 3 for cube root, etc.
3. The result and expression appear instantly.

## The formula

```
nth root of x = x^(1/n)

For x < 0 and odd n:
  result = −(|x|^(1/n))

For x < 0 and even n:
  result is complex (not real) — error shown
```

Source: Stewart J. *Calculus: Early Transcendentals.* 8th ed. §1.5

## Worked examples

**√9 = 3** — 3² = 9 (perfect square)

**√2 ≈ 1.41421356** — irrational, shown to 9 significant figures

**∛27 = 3** — 3³ = 27 (perfect cube)

**∛(−27) = −3** — (−3)³ = −27 (negative, odd root)

**⁴√16 = 2** — 2⁴ = 16 (perfect fourth root)

## Common perfect squares and cubes

| Value | Square root | Cube root |
|-------|------------|-----------|
| 1     | 1          | 1         |
| 4     | 2          | —         |
| 8     | —          | 2         |
| 9     | 3          | —         |
| 16    | 4          | —         |
| 25    | 5          | —         |
| 27    | —          | 3         |
| 64    | 8          | 4         |
| 125   | —          | 5         |

## Notes

- The root degree must be a positive whole number.
- Results are shown to 9 significant figures for irrational roots.
- The square and cube values shown are the original value squared and cubed, not the result.

## Frequently asked questions

### What is an nth root?

The nth root of x is the number r such that r^n = x. The square root is n=2, the cube root is n=3, and so on. This calculator computes x^(1/n) for any positive integer n.

### Can I take the square root of a negative number?

Not in the real numbers — the square root of a negative number is complex (imaginary). This calculator returns an error for even roots of negative values. Odd roots (like cube roots) of negative numbers are real and negative.

### What is a perfect root?

A perfect root is one where the result is an exact integer. For example, √9 = 3 (perfect square), ∛27 = 3 (perfect cube). The calculator flags these with a "perfect root" label.

### How is the nth root computed?

Using the power rule: nth root of x = x^(1/n). For negative x with odd n, the result is −(|x|^(1/n)).

### What is the cube root of a negative number?

Odd roots of negative numbers are negative real numbers. For example, ∛(−27) = −3, because (−3)³ = −27.

## Sources

- [Square root](https://en.wikipedia.org/wiki/Square_root) — Wikipedia
- [Square Root](https://mathworld.wolfram.com/SquareRoot.html) — Wolfram MathWorld

## Related calculators

- [Exponent calculator](https://onlinecalculator.me/math/exponent/) — Compute b^e for any base and exponent, including roots and negative powers.
- [Logarithm calculator](https://onlinecalculator.me/math/logarithm/) — Calculate log base 2, natural log, log base 10, or any custom base.
- [Scientific calculator](https://onlinecalculator.me/math/scientific/) — Full scientific calculator with trig, logarithms, memory, and keyboard support.
- [Percentage calculator](https://onlinecalculator.me/math/percentage/) — Five common percentage calculations — of, what percent, change, increase, decrease.

---

Part of [onlinecalculator.me](https://onlinecalculator.me/) — free calculators that run entirely in your browser.
Machine-readable index: https://onlinecalculator.me/llms.txt