Root Calculator - Square Root, Cube Root & nth Root Calculator with Step-by-Step Solutions

The number under the radical sign

The index of the root (2 for square root, 3 for cube root, etc.)

Common Roots:

  • • √16 = 4 (square root)
  • • ∛27 = 3 (cube root)
  • • 4√16 = 2 (fourth root)
  • • √(-9) = undefined (even root of negative)

Root Calculation Result

Enter values and click Calculate to see results

Root Properties & Rules

Product Rule

ⁿ√a × ⁿ√b = ⁿ√(a×b)

Multiply roots with same index

Example:

√2 × √8 = √(2×8) = √16 = 4

Quotient Rule

ⁿ√a ÷ ⁿ√b = ⁿ√(a÷b)

Divide roots with same index

Example:

√50 ÷ √2 = √(50÷2) = √25 = 5

Power Rule

(ⁿ√a)^m = ⁿ√(a^m)

Power of a root

Example:

(√2)^2 = 2
(∛8)^3 = 8

Exponent Form

ⁿ√a = a^(1/n)

Root as fractional exponent

Example:

√16 = 16^(1/2) = 4
∛27 = 27^(1/3) = 3

About Root Calculator

Our Root Calculator is a powerful and free online tool designed to calculate square roots (√), cube roots (∛), and nth roots of any number. Whether you're a student learning algebra, a professional engineer, or simply need quick root calculations, our calculator provides instant, accurate results with detailed step-by-step solutions.

Understanding roots is fundamental to mathematics, science, and engineering. A root is the inverse operation of raising a number to a power. For example, the square root of 16 is 4 because 4² = 16. Our calculator handles all types of roots and provides multiple result formats including decimal, exact radical form, and exponent notation.

Key Features

  • Square Root Calculator (√): Calculate square roots instantly with perfect square detection
  • Cube Root Calculator (∛): Compute cube roots including negative numbers
  • nth Root Calculator: Calculate any root index (4th root, 5th root, etc.)
  • Root Operations: Multiply, divide, add, and subtract roots with automatic simplification
  • Multiple Result Formats: View results in decimal, exact radical form, and exponent notation
  • Step-by-Step Solutions: Understand the calculation process with detailed explanations
  • Perfect Root Detection: Automatically identifies perfect squares, cubes, and higher powers
  • Root Simplification: Simplifies radical expressions automatically
  • Mobile-Friendly: Works seamlessly on all devices
  • 100% Free: No registration or payment required

Understanding Roots

What is a Root?

A root is a mathematical operation that finds a value which, when raised to a specific power, produces the original number. The notation ⁿ√x represents the nth root of x, where:

  • n is the root index (degree of the root)
  • x is the radicand (the number under the radical sign)
  • ⁿ√x = y means that y^n = x

Common Types of Roots

  • Square Root (√x or ²√x): The most common root. √16 = 4 because 4² = 16
  • Cube Root (∛x or ³√x): ∛27 = 3 because 3³ = 27
  • Fourth Root (⁴√x): ⁴√16 = 2 because 2⁴ = 16
  • Fifth Root (⁵√x): ⁵√32 = 2 because 2⁵ = 32

Root Properties and Rules

1. Product Rule

ⁿ√(a × b) = ⁿ√a × ⁿ√b
The root of a product equals the product of the roots.
Example: √(4 × 9) = √4 × √9 = 2 × 3 = 6

2. Quotient Rule

ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b
The root of a quotient equals the quotient of the roots.
Example: √(64 ÷ 4) = √64 ÷ √4 = 8 ÷ 2 = 4

3. Power Rule

(ⁿ√a)^m = ⁿ√(a^m)
A root raised to a power can be rewritten as the root of the radicand raised to that power.
Example: (√2)² = 2

4. Exponent Form

ⁿ√x = x^(1/n)
Any root can be expressed as a fractional exponent.
Example: √16 = 16^(1/2) = 4, ∛8 = 8^(1/3) = 2

Practical Applications

  • Geometry: Calculate side lengths of squares and cubes from their areas and volumes
  • Physics: Compute velocities, distances, and other physical quantities
  • Engineering: Structural calculations, electrical impedance, and design parameters
  • Finance: Calculate compound interest rates and investment growth factors
  • Statistics: Standard deviation and root mean square calculations
  • Computer Science: Algorithm complexity analysis and data structure operations
  • Architecture: Proportion calculations and scale factor determinations
  • Chemistry: Molecular calculations and reaction rate computations

Tips for Working with Roots

  1. Identify Perfect Powers: Recognize perfect squares (4, 9, 16, 25...) and cubes (8, 27, 64...) for quick mental calculations
  2. Simplify First: Factor the radicand to identify perfect powers before calculating
  3. Use Exponent Form: Convert roots to fractional exponents for easier algebraic manipulation
  4. Check Sign Rules: Remember that even roots of negative numbers are undefined in real numbers
  5. Rationalize Denominators: When a root appears in a denominator, multiply by the conjugate to rationalize
  6. Approximate First: Estimate the result before calculating to verify your answer makes sense

Common Root Values Reference

Perfect Squares:

√1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10

Perfect Cubes:

∛1=1, ∛8=2, ∛27=3, ∛64=4, ∛125=5, ∛216=6, ∛343=7, ∛512=8, ∛729=9, ∛1000=10

Why Use Our Root Calculator?

Our Root Calculator stands out for its accuracy, speed, and comprehensive features. Unlike basic calculators that only provide decimal results, we show multiple representations including exact radical form and exponent notation. The step-by-step solutions help you understand the calculation process, making it an excellent learning tool.

Whether you need to calculate a simple square root or perform complex root operations, our calculator handles it all with ease. The intuitive interface works perfectly on desktop and mobile devices, ensuring you can perform calculations wherever you are. Best of all, it's completely free with no registration required.