Quadratic Equation Solver (ax² + bx + c = 0)
About nth Root Calculator
The Arbitrary N-th Root ($\sqrt[n]{x} = x^{1/n}$) & Radical Exponent Calculator computes $n$-th degree roots for any real or decimal index $n$, extracting simplified radicals and plotting all $n$ complex roots evenly spaced on the complex unit circle.
How to Use nth Root Calculator
Step 1
Enter the base number (radicand $x$) and root degree ($n$).
Step 2
Review computed principal root value.
Step 3
Inspect the list of all $n$ complex solutions on the complex circle diagram.
Step 4
Click "Copy Root Value".
Practical Use Cases for nth Root Calculator
Financial Compound Interest & CAGR Exponential Roots
Compute annual growth rates by taking the $n$-th root of total capital expansion ($r = \sqrt[n]{\text{Final}/\text{Initial}} - 1$).
Acoustics, Music Temperament & Equal Octave Frequencies
Calculate 12-tone equal temperament pitch ratios ($f_{n} = f_0 \times (\sqrt[12]{2})^n$) for musical instrument tuning.
Input & Output Examples
Calculating 12th Root of 2 (Musical Semitone Ratio)
Radicand ($x$): `2`, Degree ($n$): `12`
Principal Root: `1.059463094...` | Step-by-step De Moivre theorem evaluation of all 12 complex roots
Key Features & Performance
- ✓Supports any integer or fractional root index ($n = 2, 3, 4, 5, \dots 100$ or decimal $n = 2.5$).
- ✓De Moivre Complex Root Solver: plots all $n$ complex solutions ($z_k = r^{1/n} e^{i(\theta + 2\pi k)/n}$) on the Argand diagram.
- ✓High-precision Newton-Raphson approximation engine.
- ✓100% Client-Side math engine.
- ✓1-Click Copy principal root & complex solutions.
Key Terminology & Definitions
N-th Root ($\sqrt[n]{x}$)
A number $r$ which, when raised to the power $n$, equals $x$ ($r^n = x$).
De Moivre Theorem for Roots
A trigonometric formula used to compute all $n$ distinct roots of a complex number evenly spaced around a circle by $2\pi / n$ radians.