# Free online Nth root calculator

> The nth root of a number x is the value r such that rⁿ = x. This calculator handles any integer root from 2 to 100, including negative values with odd degrees.

Calculate the nth root of any real number, from square roots to 100th roots. Shows the result, whether it is a perfect root, and step-by-step working.

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

## How to use

1. Enter the number whose root you want to find.
2. Set the degree (2 = square root, 3 = cube root, etc., up to 100).
3. The result and step-by-step working appear instantly.
4. Click "Share with my numbers" to copy a URL with your inputs.

## Formula

```
ⁿ√x = x^(1/n)
```

For negative x with odd n:
```
ⁿ√(−|x|) = −(|x|^(1/n))
```

## Root symbols

| Degree | Symbol |
|--------|--------|
| 2      | √ (square root) |
| 3      | ∛ (cube root) |
| 4      | ∜ (fourth root) |
| n ≥ 5  | ⁿ√ |

## Worked examples

- ∛64 = 4 (because 4³ = 64) — perfect root
- ∜16 = 2 (because 2⁴ = 16) — perfect root
- √2 ≈ 1.4142135624 — not a perfect root
- ∛(−8) = −2 (real, odd root of negative number)
- ⁴√(−16) — no real result (even root of negative)

## Notes

- Root degree must be an integer between 2 and 100.
- Even roots of negative numbers have no real solution.
- Results that are not perfect roots are shown to 10 significant figures.

## Frequently asked questions

### What is the nth root?

The nth root of x is the number r such that rⁿ = x. The square root is the 2nd root, the cube root is the 3rd root, and so on.

### Can I take an even root of a negative number?

No — an even root of a negative number produces a complex (imaginary) result, which this calculator does not support. Odd roots of negative numbers are real and are supported.

### What is a perfect root?

A perfect root is one where the result is an exact integer. For example, ∛64 = 4 is a perfect cube root.

### How is the nth root calculated?

The nth root of x is computed as x^(1/n). For negative x with odd n, the result is −(|x|^(1/n)).

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

For any real odd root of a negative number, the result is negative. For example, ∛(−8) = −2 because (−2)³ = −8.

## Sources

- [Nth root](https://en.wikipedia.org/wiki/Nth_root) — Wikipedia
- [nth Root](https://mathworld.wolfram.com/nthRoot.html) — Wolfram MathWorld

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