How this calculator works
This calculator finds the nth root of a number: enter the number and how many times a value must be multiplied by itself to reach it, and it returns that root. A root degree of 2 is the familiar square root, and a degree of 3 is the cube root, but any positive whole-number degree works.
Roots and exponents are inverses of each other, so the result is also checked against the property that raising the computed root back to the nth power reproduces the original number. Odd-degree roots (cube, 5th, 7th...) of a negative number are ordinary real numbers, but even-degree roots (square, 4th, 6th...) of a negative number are not real, since no real number multiplied by itself an even number of times can produce a negative result.
The formula
nth root of x: x^(1/n)Square root (n = 2): x^(1/2), written √xCube root (n = 3): x^(1/3), written ∛xFor a negative x with an odd n, this calculator computes -((-x)^(1/n)) to get the correct real result, since raising a negative number to a fractional power directly is undefined in standard floating-point math. For a negative x with an even n, no real root exists, and the calculator returns a message instead of a number.
Worked example: the cube root of -8
- Cube roots are defined for negative numbers, unlike square roots, because an odd number of negative factors multiplies to a negative result.
- Test -2: (-2)^3 = -2 × -2 × -2 = 4 × -2 = -8, which matches the target exactly.
- So the cube root of -8 is -2.
- By contrast, the square root of -8 has no real answer, since no real number squared can be negative — this calculator would show a message rather than a number for that case.
Frequently asked questions
What is the difference between a square root and a cube root?
A square root (n = 2) asks what number squared gives x; a cube root (n = 3) asks what number cubed gives x. Square roots of negative numbers are not real, while cube roots of negative numbers are ordinary real (negative) numbers.
Why can't I take an even root of a negative number?
Any real number raised to an even power is zero or positive, so no real number can square (or raise to the 4th, 6th, ...) to a negative result. The result exists only as a complex number, which this calculator does not compute.
Why does the cube root of a negative number come out negative?
An odd number of negative factors multiplies to a negative result: (-2) × (-2) × (-2) = -8. So the cube root of a negative number is simply the negative of the cube root of its positive counterpart.
Is the nth root the same as raising to a fractional exponent?
Yes — the nth root of x is defined as x raised to the power 1/n. The exponent calculator on this site computes the same operation directly if you prefer to think in terms of exponents.
What is the root degree of 1?
The 1st root of any number is the number itself, since raising a value to the power 1 leaves it unchanged.