Nth Root Calculator – x^(1/n) Any Index

The nth root generalizes square and cube roots to any index: ⁴√81 = 3 because 3⁴ = 81, and ⁵√32 = 2 because 2⁵ = 32. Odd indexes accept negative inputs — ³√−27 = −3 — while even indexes require non-negative x, since no real number raised to an even power produces a negative. The calculator verifies each result by re-exponentiating (3⁴ = 81.0000) and displays the exponent form x^(1/n) used in programming, where 81**0.25 = 3. For non-perfect values it shows Newton iteration: approximating ⁴√100 from a guess of 3, the update x ← ((n−1)x + A/xⁿ⁻¹)/n walks 3 → 3.1623 → 3.16227766, matching √10 exactly (a fourth root of 100 equals the square root of 10). Fractional powers combine rules: 8^(2/3) = (³√8)² = 4.

1. 1

   Enter the radicand x

   Any non-negative number, or negative for odd roots

2. 2

   Enter the index n

   2 for square, 3 for cube, or any higher index

3. 3

   Extract the root

   y is found so that yⁿ = x

4. 4

   Verify and convert

   Re-exponentiation and the x^(1/n) form are displayed

• 1

  ⁿ√x = x^(1/n) — the universal spreadsheet form

• 2

  Even indexes need x ≥ 0; odd indexes accept negatives

• 3

  Root of a power: ⁿ√(xⁿ) = x

• Requesting ⁴√−16, which has no real answer

• Typing x^1/n and computing (x¹)/n instead of x^(1/n)

• Assuming ⁴√81 is 9 — that is the square root; the fourth root is 3

• Losing the sign on odd roots of negatives