Square Root / Nth Root Calculator

How This Calculator Works

The nth root of a number is the value that, when raised to the nth power, gives back the original number. Enter a number and a root degree (2 for a square root, 3 for a cube root, and so on), and this calculator finds that value.

A square root (degree 2) and a cube root (degree 3) are the two most commonly used cases, but the same idea extends to any whole-number degree. A negative number only has a real nth root when the degree is odd — an even root (like an ordinary square root) of a negative number isn’t a real number at all, since no real number multiplied by itself an even number of times can ever produce a negative result.

The Formula

$$\sqrt[\vB{n}]{\vA{x}} = \vA{x}^{\frac{1}{\vB{n}}}$$

where $\vA{x}$ is the number and $\vB{n}$ is the root degree. Checking the answer is straightforward: raising the result back to the $\vB{n}$th power should reproduce $\vA{x}$.

Worked Example

1. Square root: $\sqrt{\vA{144}} = 12$, since $12^2 = 144$.
2. Cube root: $\sqrt[3]{\vA{27}} = 3$, since $3^3 = 27$.
3. Negative number, odd degree: a real answer still exists — $\sqrt[3]{\vA{-8}} = -2$, since $(-2)^3 = -8$.
4. Negative number, even degree: an even root of a negative number, like $\sqrt{-8}$, has no real answer — this calculator flags that case rather than returning a misleading result.