How this instrument works
The nth root of a number x is whichever value, raised back to the power n, returns x again — the operation is defined as the inverse of exponentiation, the same relationship that lets square root undo squaring and cube root undo cubing, only extended to any whole-number index n instead of one fixed choice. Algebraically ⁿ√x and the fractional exponent x^(1/n) name the identical operation; this sheet accepts whatever n you type rather than locking it in advance the way a dedicated square-root or cube-root tool does.
Whether a real answer exists at all turns on the parity of n. Raise any real number to an odd power and the map stays one-to-one — negative inputs stay negative, positive inputs stay positive — so every odd root, 3rd, 5th, 7th, and on, is defined for every real x and simply carries x's sign through: the cube root of −8 is −2, because (−2)³ closes the loop back to −8. Raise a real number to an even power, though, and x and −x land on the same result, so the map folds two inputs onto one output and is no longer invertible over the whole real line; an even root only undoes the non-negative half of that fold, which is why ⁿ√x for even n is defined solely for x ≥ 0 and returns the non-negative root by convention, the same rule the ordinary square root already follows.
That generality is the point of this particular sheet. The cube-root, fourth-root, and fifth-root pages elsewhere on this site each lock n at one fixed value, useful when that is the only root ever needed, while this one takes the 7th root of a number as readily as the 20th. One pattern worth noticing as n climbs: for any x above 1, ⁿ√x settles toward 1 the larger n gets, since the number that must be multiplied by itself n times to reach x has less and less room to differ from 1 as the multiplication count grows — the 100th root of 32 is already under 1.04, far tamer than the 5th root of 2.
- Enter the number you want a root of into the x field — any real number is accepted, positive or negative.
- Enter the index into the Root, n field — this sets which root is taken: 2 for square, 3 for cube, or any whole number above that.
- Read the answer in the ⁿ√x field, the sheet's Result — it updates as soon as both fields hold valid values.
- With an odd n, try a negative x to watch the sign carry straight through; with an even n, keep x at zero or above, since no real even root of a negative number exists.
Worked example — the 5th root of 32
Set x to 32 and Root, n to 5. The sheet returns exactly 2.0, because 2 multiplied by itself five times closes the loop back to 32: 2×2×2×2×2 = 32, so 2 is the number that raising to the fifth power reproduces the input untouched. Five is odd, so the formula's sign check is active even though it never has to flip anything here, since 32 is already positive.
The same sheet handles both domain cases without switching modes. With x at 81 and n at 4 — an even index — the result comes out 3.0, since 3⁴ = 81 and no sign question arises for a positive base. With x at −8 and n at 3 — an odd index — the result comes out −2.0, since (−2)³ = −8: the negative sign passes straight through instead of being rejected, exactly the behavior an odd root is supposed to have.
Questions
What does the nth root of a number actually mean?
ⁿ√x is the number that, raised to the power n, returns x — the inverse of exponentiation, generalized past square and cube roots to any whole-number index. It is identical to the fractional exponent x^(1/n); the two notations name one operation, ⁿ√x = x^(1/n), just written two different ways.
Why can't I take an even root of a negative number?
Because no real number, positive or negative, produces a negative result when raised to an even power — squaring, raising to the fourth power, and so on always land on zero or positive, so there is nothing real for an even root to undo. An odd root faces no such block, since raising a negative number to an odd power stays negative.
How does an odd root handle a negative input?
It carries the sign straight through: the cube root of −8 is −2, because (−2)³ = −8, and the same pattern holds for the 5th, 7th, or any odd root. Raising to an odd power is one-to-one across the whole real line, so every real number, negative ones included, has exactly one real odd root.
How is this different from the site's cube-root or fifth-root calculators?
Those pages fix n at one specific value — 3 for cube root, 5 for fifth root — while this sheet takes n as an input, so it reaches the 7th root, the 20th root, or any index those dedicated pages don't cover. Use a fixed-n page when that is the only root ever needed; use this one when n itself changes from problem to problem.
What is the relationship between ⁿ√x and a fractional exponent?
They are the same number written two ways: ⁿ√x = x^(1/n). The broader identity x^(m/n) = (ⁿ√x)^m builds directly on this equivalence, taking the root first and then raising that result to a further power m.
Does the answer ever come out as an irrational number?
Yes, almost always — most integers are not perfect nth powers, so most roots are irrational and the field shows a rounded decimal rather than an exact value. 32 happens to equal 2⁵ exactly, so its 5th root lands on a clean 2.0; the 5th root of 30, by contrast, never terminates or repeats.