Root Calculator
Calculate the nth root of any number, including square roots, cube roots, and negative or reciprocal root indices, with clear rules for negative radicands.
How It's Calculated
Formula
\sqrt[n]{x} = x^{\frac{1}{n}}The nth root of a number x is the value that, multiplied by itself n times, gives back x. The square root (n=2) and cube root (n=3) are the most common cases, but any whole-number root index works the same way: it's equivalent to raising x to the fractional power 1/n. A positive radicand has a real nth root for any root index. A radicand of zero has a root of zero, as long as the index is positive — a negative index on a zero radicand is undefined, since it's equivalent to dividing by zero. A negative radicand only has a real-valued root when the root index is odd: the cube root of -8 is -2, because -2 x -2 x -2 = -8. An even root index (square root, fourth root, and so on) of a negative number has no real-valued result, since no real number multiplied by itself an even number of times produces a negative result. A negative root index represents a reciprocal root: x^(1/-n) equals 1 divided by x^(1/n), so the square root of 16 with index -2 is 1/4.
Worked Examples
Square root of 16 (index 2)
- 16^(1/2) is the value that, squared, gives 16
- 4 x 4 = 16
- Result: 4
Cube root of a negative number: -8 (index 3)
- -8^(1/3) is the value that, cubed, gives -8
- Since 3 is odd, a negative radicand has a real result
- -2 x -2 x -2 = -8
- Result: -2
Frequently Asked Questions
Why does an even root of a negative number have no real result?
Multiplying any real number by itself an even number of times always gives a non-negative result — a negative number times itself is positive, and that pattern holds for any even count of multiplications. So no real number can be squared, fourth-rooted, or otherwise even-rooted to produce a negative radicand, which is why this calculator rejects that combination instead of returning an invalid value.
Why does an odd root of a negative number work?
An odd number of negative factors multiplies out to a negative result (for example, -2 x -2 x -2 = -8, three negative factors). So a negative radicand always has a real-valued root when the root index is odd, and this calculator applies the sign explicitly to compute it correctly, rather than relying on the JavaScript `Math.pow` function, which returns an invalid result for a negative base with a fractional exponent even when the true root is real.
What does a negative root index mean?
A negative root index represents the reciprocal of the corresponding positive-index root. x^(1/-n) equals 1 / x^(1/n). For example, 16 with root index -2 first computes the square root of 16 (which is 4), then takes the reciprocal: 1/4.
Why is a zero radicand with a negative root index invalid?
A negative root index means taking a reciprocal, and the reciprocal of 0 (1/0) is undefined — it's a division by zero. So a zero radicand is only valid with a positive root index, where the result is simply 0.