Root Calculator: Square, Cube & Nth Root Solver
Free square root calculator, cube root calculator, and nth root solver. How to find square root and nth root formula with steps. Find ⁿ√a or solve for radicand (inverse). Newton's method, verification bⁿ ≈ a. Trusted by students. No sign-up.
What This Root Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:Root from degree + radicand; inverse: radicand from degree + result. Perfect Root highlight, verification bⁿ ≈ a, Logic Trace (exponential form a^(1/n)), Newton's method Estimation table (3 iterations), Copy results, rounding/sig figs.
- Ideal Users:Students: How to find square root, cube root, nth root formula. Teachers: Newton's method demo, exponential form. Anyone: Quick square root calculator or 8th root calculator for homework or verification.
- Scope:Degree n: positive integer. Even roots of negative numbers undefined. All calculations local.
This root calculator computes nth roots (ⁿ√a = b) and supports square root, cube root, and general roots (4th, 5th, 8th, etc.). Inverse mode solves for the radicand when you know the root.
Nth Root Formula: ⁿ√a = a^(1/n)
The nth root of a is b where
In exponential form:
Examples: √16 = 4, ∛27 = 3, ⁸√15 ≈ 1.403. The Logic Trace substitutes your values into this form
Square Root, Cube Root & General Nth Root
Square root (n=2) and cube root (n=3) have dedicated presets. For nth root calculation with n = 4, 5, 8, etc., enter the degree directly. The radical notation ⁽ⁿ⁾√a updates dynamically.
Inverse Mode: Solve for Radicand
If you know the root (b) and degree (n), use Inverse Mode to find the radicand: a = bⁿ. Example: ⁸√a = 1.5 → a = 1.5⁸ ≈ 25.63. Enter Degree and Result; leave Radicand blank.
Newton's Method: How Nth Roots Are Approximated
The Root Lab uses Newton's method, a standard numerical algorithm with quadratic convergence:
The Estimation Table shows the first 3 iterations—useful for understanding how the approximation converges
Root Calculator FAQ
? What is an nth root?
The nth root of a is the number b such that bⁿ = a. Notation: ⁿ√a = b. Square root (n=2): √16 = 4. Cube root (n=3): ∛27 = 3. This calculator finds square roots, cube roots, and any general nth root (4th, 5th, 8th, etc.).
? How do I find the square root or cube root?
Use the √ Square Root (n=2) or ∛ Cube Root (n=3) presets, or set Degree manually. Enter the Radicand and leave Result blank. The verification card confirms bⁿ ≈ a.
? What is Inverse Mode?
When you know the root (b) and degree (n), Inverse Mode solves for the radicand: a = bⁿ. Enter Degree and Result; leave Radicand blank. The solved field shows a dashed mint border.
? What does "Perfect Root" mean?
A perfect root is an exact integer result (e.g. √16 = 4, ∛27 = 3). The dashboard highlights "Perfect Root" when the computed value has no decimal part.
? What is Newton's method in the Estimation Table?
The calculator uses Newton's method—a standard numerical algorithm for nth roots:
The Estimation Table shows the first 3 iterations so you see convergence
? Can I calculate roots like the 8th root or 5th root?
Yes. Enter any positive integer degree (4, 5, 8, etc.). A "General n-th Root Mode" badge appears when n > 3. Example: ⁸√15 ≈ 1.403.