Root Calculator

Square root, cube root, or any nth root of a number

Finding the square root, cube root, or any nth root of a number is the direct inverse operation of raising a number to that same power — and while square roots are the most familiar case, the same underlying concept extends to any root, including roots of negative numbers, which behave differently depending on whether the root index is even or odd.

Why odd and even roots of negative numbers behave differently

For a positive number, any nth root (square, cube, or otherwise) is well-defined and straightforward. For a negative number, the behavior depends specifically on whether the root index is even or odd: an odd root (cube root, fifth root, etc.) of a negative number is well-defined and produces a real negative result, since a negative number multiplied by itself an odd number of times stays negative. An even root (square root, fourth root, etc.) of a negative number, however, has no real-number solution — no real number multiplied by itself an even number of times can produce a negative result, which is exactly why square roots of negative numbers require complex numbers to express at all, the same territory touched on by negative discriminants in the Quadratic Formula Calculator.

A worked example

The cube root of a negative number produces a real negative result, while the square root of the identical negative number has no real-number answer at all — this isn't a limitation of calculation technique, it's a genuine mathematical fact about how even versus odd powers behave with negative bases, and it's why "the square root of a negative number" and "the cube root of a negative number" are fundamentally different kinds of questions.

How this connects to your other math calculations

Roots and exponents are inverse operations of each other — the Exponent Calculator covers the forward direction (raising to a power, including fractional powers that are themselves equivalent to roots), while this calculator covers the reverse direction specifically.

Common mistakes

Assuming every number has a real nth root regardless of sign and root index — attempting to find a real square root of a negative number, for instance — overlooks the even/odd distinction that determines whether a real solution exists at all; always check whether the root index is even or odd before expecting a real-number answer for a negative input.