Nth Root Calculator

ⁿ√x = x^(1/n). Generalizes square and cube roots to any positive integer n.

Inputs

Result

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General calculation reads

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How Not to Be Wrong
Jordan Ellenberg - math thinking
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The Joy of X
Steven Strogatz - accessible math
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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter x.
  • Enter root degree n.
  • Read decimal value.

About this calculator

The n-th root of x is the value v where v^n = x. ⁵√32 = 2 (because 2⁵ = 32). For odd n, real n-th roots exist for negatives: ⁵√(−32) = −2. For even n, only non-negative inputs have real roots — ⁴√(−16) is imaginary. Compute via x^(1/n). Used in geometry (cube → ∛volume = side), finance (annualized return from a multi-period total), and decibel-related calculations.

Frequently asked

Why does even n need non-negative x?+
Squaring (or any even power) always gives a non-negative result, so reversing it from a negative is impossible in real numbers.
Annualized return from compound growth?+
If your money grew 60% over 4 years, annualized rate = ⁴√1.60 − 1 ≈ 0.124 ≈ 12.4%/year.
Difference between √, ∛, ⁿ√?+
All special cases. √x = x^(1/2). ∛x = x^(1/3). ⁿ√x = x^(1/n). Same operation, different n.
Is it accurate?+
IEEE 754 ~15 sig figs. For perfect roots, may show a tiny floating-point error (e.g. 1.9999998).
How fast?+
Math.pow uses log+exp; very fast. For arbitrary precision, Newton iteration on integer inputs.

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