Nth Root Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
The nth root of a number x is calculated as x^(1/n). The cube root of 27 is exactly 3, since 3³ = 27.
Explanation
The nth root of a number x is the number which, raised to the power n, gives back x: the square root (n=2) is the best known, the cube root (n=3) the second most common, but the principle generalizes to any integer index. Mathematically, extracting an nth root is equivalent to raising the number to the power 1/n, an operation directly linked to standard exponentiation and its usual calculation rules. This calculator deliberately limits itself to non-negative numbers: for a negative number with an odd root index (like the cube root of −8, which equals exactly −2 in real mathematics), the standard x^(1/n) calculation, used by nearly all programming languages and calculation tools, doesn't return a defined result for a negative base with a fractional exponent, even when the real root exists mathematically. This widespread technical limitation is documented explicitly rather than being a source of a silently incorrect result — the same discipline applied to the cap on our factorial calculator. For a positive number, on the other hand, the nth root always exists and this calculator gives it directly, whether it's an exact integer (like the cube root of 27) or an irrational number for which only an approximate value can be displayed (like the square root of 2, about 1.414214). This operation regularly comes up in other mathematical calculations, for example to isolate a variable in our quadratic equation calculator, which directly involves a square root in its solving formula.
Example: cube root of 27
Inputs
Number: 27. Root index: 3.
Calculation
27^(1/3) = 3, since 3 × 3 × 3 = 27.
Result
The cube root of 27 is exactly 3.
Frequently asked questions
Why doesn't this calculator accept negative numbers?
For an even root index (square root, 4th root...), no real root exists for a negative number (the square of a real number is never negative). For an odd index (cube root...), a real root does exist mathematically for a negative number, but the standard x^(1/n) operation used by this calculator, like most calculation tools, doesn't compute it correctly for a negative base: it returns an undefined result rather than the expected real value.
How do I mentally calculate the cube root of a negative number?
For a negative number and an odd root index, calculate the root of the corresponding positive number (ignoring the sign), then restore a negative sign to the final result: the cube root of −8, for example, is −2, exactly the opposite of the cube root of 8 (which is 2). This rule only applies to odd root indices.
What's the difference between an exact root and an approximate root?
A root is exact when the result is a whole number or a simple fraction (like the cube root of 27, exactly 3); it's approximate when the result is an irrational number, which can only be represented by a truncated decimal value (like the square root of 2, whose decimals continue indefinitely with no repeating pattern). This calculator displays a decimal value precise to six digits after the decimal point in both cases.