Compute the square root, cube root, or any nth root.

What "nth root" means, and the default worked

The nth root of x is the number that, raised to the power n, gives back x. This calculator computes it as x^1∕n, which is why an nth root and an nth power undo each other — they are the same operation with the exponent flipped. Its own default, the square root (n=2) of 16, returns 4, since 4² = 16.

Negative numbers: sometimes fine, sometimes not

A negative number has a real cube root because a negative number cubed is negative — (−2)³ = −8, so the cube root of −8 is −2, which is what this calculator returns. Any odd root of a negative number works the same way. An even root cannot: raise any real number to an even power and the result is never negative, so no real number squared, to the 4th, or to any even power produces a negative x, and there is no real answer to return. The calculator flags this rather than guessing — the honest answer is that one does not exist within real numbers, not that the arithmetic failed.

A rounding quirk worth knowing about, not worrying about

Computing a root as x^1∕n runs into an unavoidable quirk of how computers store fractions: 1∕3 cannot be represented exactly in binary floating point, so the cube root of 64 computes internally as 3.9999999999999996, not a clean 4, even though 4³ is exactly 64. This is not a bug in this calculator — every calculator and programming language built on standard floating-point arithmetic hits the identical rounding. It only matters if you read out fifteen decimal places; rounded to six, as this page does, it correctly shows 4.000000.

The inverse of the exponent calculator

Roots and exponents are the same relationship read in opposite directions: the exponent calculator asks "what is x raised to the power n", and this one asks "what, raised to the power n, gives x". Fractional exponents make the connection explicit — raising a number to the power 1∕2 is taking its square root, so the exponent calculator can compute any root you need here simply by entering 1∕n as the exponent, and the two tools will always agree.

What is an nth root?

The nth root of x is the number that, when raised to the power n, gives x back. The square root (n=2) of 16 is 4 because 4² = 16; the cube root (n=3) of 27 is 3 because 3³ = 27. Higher roots follow the same pattern with a larger index.

Why can't you take an even root of a negative number (in real numbers)?

Because any real number raised to an even power always comes out positive or zero — a negative number multiplied by itself an even number of times cancels its own negative sign out. So there's no real number that, squared or raised to any even power, produces a negative result, meaning an even root (square root, 4th root, etc.) of a negative number has no answer within the real numbers — only within the complex number system.

How is a cube root of a negative number possible?

Odd roots preserve the sign of the original number, unlike even roots. The cube root of −8 is −2, because (−2)³ = −2 × −2 × −2 = −8 — the negative sign survives an odd number of multiplications. This holds for any odd root of a negative number, so this calculator returns a real, negative result whenever the index is odd and the value is negative.

How is this different from your Exponent Calculator?

A root is the inverse operation of raising to a power: the nth root of x equals x^(1/n), which is exactly what the Exponent Calculator computes when you give it a fractional exponent of 1/n. This tool simply frames the same operation the way roots are usually taught — value and root index — while the Exponent Calculator frames it as base and exponent. Both tools agree on the underlying math.

Worked example: the square root of 16 is 4 (n=2, since 4² = 16). The cube root of −8 is −2 (n=3, an odd index, so the negative value has a real answer). Attempting the square root of −8 instead (n=2, an even index) has no real answer — this calculator flags that case with a warning rather than showing a misleading number.