Square Root Calculator
Calculate square roots (√x) and N-th root radical expressions.
GCD & LCM Calculator
About Square Root Calculator
The Square Root & Nth Radical Calculator computes Principal Square Roots ($\sqrt{x}$), Cube Roots ($\sqrt[3]{x}$), arbitrary $N$th Roots ($\sqrt[n]{x}$), and simplified radical forms ($a\sqrt{b}$) with Babylonian iterative steps.
How to Use Square Root Calculator
Step 1
Enter the radicand number ($x$) and root degree ($n$, default 2 for square root).
Step 2
Click "Calculate Root".
Step 3
Review exact decimal value and simplified radical form ($a\\sqrt{b}$).
Step 4
Click "Copy Result".
Practical Use Cases for Square Root Calculator
Geometry & Euclidean Distance Calculations
Compute hypotenuses via the Pythagorean theorem ($c = \sqrt{a^2 + b^2}$) and diagonal distances across 2D/3D coordinate spaces.
Simplifying Algebraic Radicals
Simplify square roots of non-perfect squares into exact mixed radical form (e.g. $\sqrt{72} = 6\sqrt{2}$).
Input & Output Examples
Simplifying Radical of 72
Number: 72, Root Degree: 2 (Square Root)
Principal Square Root: 8.485281 | Simplified Radical: $6\\sqrt{2}$ | Prime Factors: $2^3 \\times 3^2$Key Features & Performance
- ✓Computes Square Root, Cube Root, and arbitrary $N$th Root ($\\sqrt[n]{x}$).
- ✓Auto-simplifies non-perfect squares into exact simplified radical form ($a\\sqrt{b}$).
- ✓Supports high-precision decimal expansion up to 50 decimal places.
- ✓100% Client-Side calculation.
- ✓1-Click Copy radical result.
Key Terminology & Definitions
Principal Square Root
The unique non-negative real number that, when multiplied by itself, equals the given non-negative radicand.
Simplified Radical Form
An expression written as $a\\sqrt{b}$ where the radicand $b$ contains no perfect square factors other than 1.