Root Calculator

Calculate square roots, cube roots, and any nth roots with precision

x
√
n √x = r

where rⁿ = x

Root Formulas

Square Root: √x = r where r² = x

Cube Root: ³√x = r where r³ = x

Nth Root: ⁿ√x = r where rⁿ = x

  • Properties of Roots:
    • ⁿ√(a × b) = ⁿ√a × ⁿ√b
    • ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b
    • (ⁿ√a)ᵐ = ⁿ√(aᵐ)
    • ᵐ√(ⁿ√a) = ᵐⁿ√a
Common Roots
Number Square Root Cube Root
1 1 1
4 2 ≈1.587
9 3 ≈2.080
16 4 ≈2.520
25 5 ≈2.924
Multiple Root Types

Calculate square roots, cube roots, or any nth root with our versatile calculator.

Step-by-Step Solutions

Understand the calculation process with detailed step-by-step explanations.

Adjustable Precision

Control decimal places from 0 to 8 for your specific needs.

About the Root Calculator

Our Root Calculator is a powerful mathematical tool designed to help students, teachers, engineers, and professionals quickly and accurately calculate roots of numbers. Whether you need a simple square root or a complex nth root calculation, this tool provides instant results with your desired level of precision.

How to Use the Root Calculator

  1. Enter the number you want to find the root of in the "Number" field
  2. Select the type of root you want to calculate (square, cube, or nth root)
  3. If calculating nth root, enter the root value (e.g., 4 for fourth root)
  4. Choose your desired decimal precision
  5. Click "Calculate Root" to see the result with detailed steps

Understanding Roots in Mathematics

In mathematics, a root of a number x is a number r that, when raised to some positive integer power n, equals x. The most common roots are:

  • Square Root (√x or ²√x): The number that when multiplied by itself equals x
  • Cube Root (³√x): The number that when multiplied by itself three times equals x
  • Nth Root (ⁿ√x): The number that when multiplied by itself n times equals x

Roots are the inverse operation of exponents. For example, since 3² = 9, then √9 = 3.

Practical Applications of Roots

Root calculations have numerous real-world applications:

  • Geometry: Calculating side lengths from areas or volumes
  • Physics: Solving equations involving inverse-square laws
  • Engineering: Determining dimensions in design calculations
  • Finance: Calculating compound interest and growth rates
  • Computer Science: Algorithm complexity analysis
  • Statistics: Calculating standard deviations

Example Problems

Example 1: Square Root

What is the square root of 64?

Solution:

√64 = 8 because 8 × 8 = 64

Example 2: Cube Root

What is the cube root of 125?

Solution:

³√125 = 5 because 5 × 5 × 5 = 125

Example 3: Fourth Root

What is the fourth root of 81?

Solution:

⁴√81 = 3 because 3 × 3 × 3 × 3 = 81

Example 4: Decimal Root

What is the square root of 2 to 4 decimal places?

Solution:

√2 ≈ 1.4142