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.
Disclaimer: This calculator performs standard mathematical operations for general and educational use only. We are not a substitute for professional review — for calculations where accuracy is critical, such as financial filings, engineering work, or academic submissions, please verify the result independently.
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.