Nth Root Calculator
Compute the nth root (ⁿ√x), invert to a power, or solve for the root index, with exact and decimal forms and step-by-step simplification.
Nth root
Updates as you typeMode
What do you want to solve for? ?
Quick values (click to fill)
Common x
Common n
Inputs
Radicand (x) ?
—
Root index (n) ?
—
even
Result value (y) ?
—
Display (optional)
Decimal precision ?
Formula
=
x1/n
Nth root: n√x = x1/n
Simplification: n√(aⁿ × b) = a · n√b
Even root: for negative x, n√x is imaginary when n is even.
Odd root: n√(−x) = −n√x when n is odd (real result).
Product rule: n√(a·b) = n√a · n√b (a, b ≥ 0)
Reciprocal root: x1/−n = 1 / x1/n
Find n: if yⁿ = x, then n = log(x) / log(y).
- x
- Radicand — the number under the radical sign.
- n
- Root index. 2 = square root, 3 = cube root, 4 = 4th root, etc.
- x1/n
- Equivalent exponential form. Often easier in algebra.
- Parity
- Whether n is even or odd — determines how negative x behaves.
Worked example — your numbers
- Radicand: —
- Root index: — (—)
- Setup: —
- Pull out the power: —
- Exact: —
- Decimal: —
As n grows, n√x approaches 1 for any positive x — the bigger the root, the smaller the effect. Negative radicands with even n produce pure imaginary results; with odd n they stay real and flip sign.