Root Calculator
Calculate square roots, cube roots, and any nth root instantly — with step-by-step formula display.
Square Root Calculator
√x = x^(1/2)
Cube Root Calculator
∛x = x^(1/3)
General Root Calculator (nth Root)
ⁿ√x = x^(1/n)
The Mathematics Behind Roots
The square root of x is the number y such that y² = x. Mathematically, √x = x¹ᐟ². Every positive real number has two real square roots — one positive (the principal root) and one negative. For example, √25 = ±5, though by convention we return the positive root.
The cube root of x is the number y such that y³ = x. Written as ∛x = x^(1/3). Unlike square roots, cube roots are defined for all real numbers including negatives, because a negative number multiplied by itself three times remains negative: (−2)³ = −8, so ∛(−8) = −2.
The nth root of x is the number y such that yⁿ = x. Computed as ⁿ√x = x^(1/n). For even values of n, the number inside the root must be non-negative (real result). For odd values of n, negative numbers are allowed. For example: ⁴√81 = 3, and ⁵√(−32) = −2.
Frequently Asked Questions
About this calculator
This root calculator lets you compute square roots, cube roots, and any nth root with a clean, step-by-step display. Each result shows the underlying formula so you can verify exactly how the answer was reached — useful for homework, engineering, or any time you need to understand the exponent relationship behind a root.
- Square root — Computes √x for any non-negative number, returning the principal (positive) root.
- Cube root — Computes ∛x for any real number, including negatives — ∛(−8) = −2.
- Nth root — Computes ⁿ√x for any positive integer or decimal root index n, with validation for even-index negatives.
- Step-by-step formula — Each result displays the full formula used, e.g. √25 = 25^(1/2) = 5, so you see exactly how the answer was derived.
- Clear error messages — Explains why a result isn't real — e.g. even-index roots of negative numbers — instead of just showing an error code.