Root Calculator

Calculate square root, cube root, and nth root

Free online root calculator for calculating square roots (√), cube roots (∛), and nth roots. Real-time calculation with instant results. No registration required.

How to Use

Simple 3-step process to calculate roots:

Step 1: Enter Root Degree
Enter the root degree (n). For square root use 2, for cube root use 3, or any positive integer for nth root
Step 2: Enter Number (Radicand)
Enter the number under the root (radicand). Can be any real number (negative for odd roots only)
Step 3: Get Result
The result is calculated instantly in real-time. Use quick calculation buttons for common roots

All calculations are performed in your browser. No data is sent to our servers.

Root Calculator Tool

Root Degree (n) 2 = square root, 3 = cube root

√

Number (Radicand) The number under the root symbol

Quick Calculations

Result

Practical Use Cases

Root calculators are essential in various fields:

1. Geometry and Area Calculations

Calculate side lengths from areas. If a square has area 25 m², the side length is √25 = 5m. If a cube has volume 64 m³, the side length is ∛64 = 4m. Useful for construction, architecture, and spatial planning.

2. Physics and Engineering

Root calculations in formulas. Velocity: v = √(2gh). RMS voltage: Vrms = √(V₁² + V₂²). Standard deviation: σ = √(Σ(x-μ)²/n). Essential for electrical engineering, mechanics, and statistics.

3. Finance and Investment

Calculate compound annual growth rate (CAGR): CAGR = (Final/Initial)^(1/years) - 1. If investment doubles in 10 years, growth rate = ¹⁰√2 - 1 ≈ 7.18% per year. Used in portfolio analysis and return calculations.

4. Computer Graphics and Game Development

Distance calculations using Pythagorean theorem: distance = √(x² + y² + z²). Normalize vectors: v/√(v₁² + v₂² + v₃²). Calculate lighting intensity and collision detection in 3D graphics.

5. Mathematics Education

Learn root properties and relationships. √4 = 2, √9 = 3, √16 = 4. ∛8 = 2, ∛27 = 3, ∛64 = 4. Understanding roots is fundamental for algebra, calculus, and advanced mathematics.

What is Root Calculation?

A root is the inverse operation of exponentiation. Finding the nth root of a number means finding a value that, when raised to the power n, equals the original number.

Types of Roots

Square root (√, n=2): Most common root. √16 = 4 because 4² = 16. Cube root (∛, n=3): Third root. ∛27 = 3 because 3³ = 27. Nth root (ⁿ√, general): Any positive integer n. ⁴√81 = 3 because 3⁴ = 81. Each type has specific applications in mathematics and science.

Root Properties

Positive numbers always have positive roots. √25 = 5 (principal root). Negative numbers have real odd roots: ∛(-8) = -2. Negative numbers have no real even roots: √(-4) is imaginary (2i). Zero root is always zero: √0 = 0, ∛0 = 0. Root of 1 is always 1: √1 = 1, ∛1 = 1.

Calculation Method

This calculator uses the exponential formula: ⁿ√x = x^(1/n). For example, ³√8 = 8^(1/3) = 2. Negative radicands with odd roots: ³√(-8) = -(8^(1/3)) = -2. Results are calculated with high precision (up to 10 decimal places) and trailing zeros are removed for clarity.

Frequently Asked Questions

How do I calculate a square root?

Enter 2 as the root degree and your number as the radicand. For example, to find √16, enter degree=2 and radicand=16. Result: 4. Square root is the most common root operation, used to find side length from area.

How do I calculate a cube root?

Enter 3 as the root degree and your number as the radicand. For example, to find ∛27, enter degree=3 and radicand=27. Result: 3. Cube root is used to find side length from volume.

Can I calculate roots of negative numbers?

Yes, but only for odd roots. ∛(-8) = -2 (valid). √(-4) gives an error because even roots of negative numbers are complex (imaginary). Odd roots preserve the sign of the radicand.

What is the difference between √x and x^(1/2)?

They are mathematically equivalent. √x is radical notation (traditional). x^(1/2) is exponential notation (used in calculators). Both represent the same operation: finding the number that when squared equals x.

How do I calculate higher-order roots like fourth root?

Enter the desired root degree. For fourth root of 16: degree=4, radicand=16, result=2 (because 2⁴=16). For fifth root of 32: degree=5, radicand=32, result=2 (because 2⁵=32). Any positive integer root degree is supported.

Why does √2 give a decimal instead of a fraction?

√2 is an irrational number (≈1.414213562...). It cannot be expressed as a simple fraction. The decimal representation continues infinitely without repeating. This calculator shows up to 10 decimal places for precision.

What are the quick calculation buttons for?

Quick buttons provide instant calculations for common roots: √4=2, √9=3, √16=4, √25=5 (perfect squares). ∛8=2, ∛27=3, ∛64=4, ∛125=5 (perfect cubes). These help verify calculator accuracy and learn root patterns.

How accurate is this root calculator?

This calculator uses JavaScript's Math.pow() function with double precision (about 15-17 significant digits). Results are formatted to 10 decimal places and trailing zeros are removed. For most practical purposes, this precision is more than sufficient.