Root Calculator – Square Root, Cube Root, and Nth Root Calculator

  • Adam
  • November 4, 2025

Free online root calculator for computing square roots, cube roots, nth roots, and simplifying radical expressions. Calculate √x, ∛x, ⁿ√x with step-by-step solutions and conversion to exponential form.

Root Calculator

Advanced Tool for Computing Square Roots, Cube Roots, and Nth Roots

Quick Navigation

Square Root

Calculate √x

Cube Root

Calculate ∛x

Nth Root

Calculate ⁿ√x (root of any degree)

Simplify Square Root

Simplify √x to simplest radical form

Convert to Exponential Form

Express ⁿ√x as fractional exponent

What are Roots?

A root is the inverse operation of exponentiation. If we say "2 raised to the power 3 equals 8" (2³ = 8), the root equivalent is "the cube root of 8 is 2" (∛8 = 2). Roots extract the base from a power equation.

The most common roots are square roots (√) and cube roots (∛), but roots can be of any degree (nth root). Roots appear everywhere in mathematics: geometry uses roots for the Pythagorean theorem, physics uses them for calculating distances and velocities, and statistics uses them for standard deviation calculations.

Understanding roots is crucial for algebra, calculus, and scientific work. Roots can be rational (like √4 = 2) or irrational (like √2 ≈ 1.414). This calculator helps you compute roots, simplify radical expressions, and convert between different representations.

Key Concept: Roots and exponents are inverse operations. If √x = y, then y² = x. They undo each other.

Key Features & Capabilities

This comprehensive root calculator provides multiple calculation modes and detailed analysis:

√ Square Root Calculate √x with instant results and verification
∛ Cube Root Calculate ∛x including negative number roots
ⁿ Nth Root Calculate any root of any degree (4th, 5th, etc.)
📉 Simplify Radicals Reduce square roots to simplest radical form
📊 Exponential Conversion Express roots as fractional exponents
📋 Step-by-Step Solutions See detailed breakdown showing how calculations are performed
✓ Multiple Formats Display results as decimals, fractions, and radicals
🔄 Perfect Square Detection Identifies perfect squares and simplification factors
📋 Copy to Clipboard One-click copy functionality to transfer results
🎓 Educational Content Comprehensive guides and examples
⚡ Real-Time Calculation Instant results with no delays
📱 Fully Responsive Works seamlessly on desktop, tablet, and mobile

How to Use This Calculator

Step-by-Step Guide

  1. Choose Root Type: Select the appropriate tab: Square Root, Cube Root, Nth Root, Simplify Radical, or Convert to Exponent.
  2. Enter Your Value: Input the number for which you want to find the root. For nth roots, specify both the number and the degree.
  3. Click Calculate: Press the Calculate button to perform the computation.
  4. Review Results: The main result displays prominently showing the root value.
  5. Study Steps: See detailed breakdown showing the calculation method.
  6. Analyze Statistics: View related values like simplified forms, decimal approximations, or verification.
  7. Copy or Clear: Use Copy to transfer results. Use Clear to reset for a new calculation.

Tips for Accurate Use

  • Positive Numbers: For square roots and even-degree roots, input positive numbers (or zero). Odd-degree roots accept negative numbers.
  • Exact vs Approximate: Perfect squares give exact integer results. Non-perfect squares give approximations.
  • Simplification: Always look for perfect square factors to simplify radical expressions.
  • Verification: Check your answer: if √9 = 3, then 3² = 9 should be true.
  • Degree Specification: For nth roots, make sure the degree is at least 2.

Complete Formulas Guide

Square Root

Basic Square Root Definition
If x² = y, then √y = x

Example: √16 = 4 because 4² = 16
Example: √25 = 5 because 5² = 25

Properties:
√(x × y) = √x × √y (product rule)
√(x/y) = √x / √y (quotient rule)

Cube Root

Basic Cube Root Definition
If x³ = y, then ∛y = x

Example: ∛8 = 2 because 2³ = 8
Example: ∛27 = 3 because 3³ = 27

Note: Cube roots can be taken of negative numbers:
∛(-8) = -2 because (-2)³ = -8

Nth Root

General Nth Root Definition
If x^n = y, then ⁿ√y = x

Or as an exponent: ⁿ√y = y^(1/n)

Example: ⁴√16 = 2 because 2⁴ = 16
Example: ⁵√32 = 2 because 2⁵ = 32

Radical Simplification

Simplifying Square Roots
√(a² × b) = a√b

Example: √72 = √(36 × 2) = √36 × √2 = 6√2
Example: √48 = √(16 × 3) = √16 × √3 = 4√3

Process: Find perfect square factors and extract them

Fractional Exponents

Converting Between Radicals and Exponents
ⁿ√x = x^(1/n)
ⁿ√(x^m) = x^(m/n)

Example: √8 = 8^(1/2)
Example: ∛(x²) = x^(2/3)
Example: ⁴√16 = 16^(1/4) = 2

Root Properties and Rules

Product Property

√(xy) = √x × √y - The root of a product equals the product of the roots. This is used to simplify radicals.

Quotient Property

√(x/y) = √x / √y - The root of a quotient equals the quotient of the roots. Used to simplify fractions under radicals.

Power Property

(ⁿ√x)^n = x - Taking a root and raising to the same power returns the original number. Roots and powers undo each other.

Reciprocal Property

ⁿ√x = x^(1/n) - Roots can be expressed as fractional exponents. Very useful for advanced calculations.

Combining Roots

ⁿ√x × ⁿ√y = ⁿ√(xy) - Roots with the same degree can be combined by multiplying the values under the radicals.

Worked Examples

Example 1: Simple Square Root

Problem: Calculate √64

Solution:
We need to find: x where x² = 64
√64 = 8

Verification: 8² = 8 × 8 = 64 ✓

Example 2: Cube Root with Negative

Problem: Calculate ∛(-125)

Solution:
We need to find: x where x³ = -125
∛(-125) = -5

Verification: (-5)³ = (-5) × (-5) × (-5) = -125 ✓
Note: Odd-degree roots can handle negative numbers

Example 3: Nth Root

Problem: Calculate ⁴√256

Solution:
We need to find: x where x⁴ = 256
⁴√256 = 4

Verification: 4⁴ = 4 × 4 × 4 × 4 = 256 ✓

Example 4: Simplifying a Radical

Problem: Simplify √72

Solution:
Find perfect square factors of 72:
72 = 36 × 2
√72 = √(36 × 2) = √36 × √2 = 6√2

Result: 6√2 ≈ 8.485

Example 5: Fractional Exponent Conversion

Problem: Express ∛(x²) as a fractional exponent

Solution:
Radical form: ∛(x²)

Using ⁿ√(x^m) = x^(m/n):
n = 3 (cube root)
m = 2 (power of x)

Result: x^(2/3)

Verification: (x^(2/3))³ = x^(2/3 × 3) = x² ✓

Frequently Asked Questions

❓ Is √4 equal to ±2 or just 2?
The square root symbol (√) refers to the principal (positive) square root only. So √4 = 2, not ±2. However, the equation x² = 4 has two solutions: x = 2 and x = -2.
❓ Can you take the square root of a negative number?
In real numbers, no. Square roots of negative numbers are undefined in the real number system. However, in complex numbers, √(-1) = i (the imaginary unit).
❓ Can you take the cube root of a negative number?
Yes! Cube roots of negative numbers are defined. ∛(-8) = -2 because (-2)³ = -8. Odd-degree roots (3rd, 5th, etc.) can handle negative numbers.
❓ What is a perfect square?
A perfect square is a number that equals an integer squared. Examples: 1 (1²), 4 (2²), 9 (3²), 16 (4²), 25 (5²). Perfect squares have exact integer square roots.
❓ Why simplify radicals?
Simplifying radicals makes expressions clearer and easier to work with. 6√2 is simpler and clearer than √72. It's also standard mathematical form for exact answers.
❓ What is an irrational root?
An irrational root cannot be expressed as a simple fraction. Examples: √2 ≈ 1.41421..., √3 ≈ 1.73205..., √5 ≈ 2.236... These go on forever without repeating.
❓ How are roots used in geometry?
The Pythagorean theorem uses roots: c = √(a² + b²). This finds the hypotenuse of a right triangle. Roots also calculate distances and lengths in geometric problems.
❓ What is the relationship between roots and fractional exponents?
They're equivalent: √x = x^(1/2), ∛x = x^(1/3), ⁿ√x = x^(1/n). This lets you use exponent rules on radical expressions, making complex calculations easier.
❓ When would I use this calculator?
Use for: solving equations with radicals, simplifying expressions, geometry problems, physics calculations, engineering work, statistics (standard deviation), computer science (square root algorithms), or any math involving roots.

Start Calculating Roots

Whether you're solving algebraic equations, working with geometry, studying advanced mathematics, or analyzing scientific data, this comprehensive root calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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