Root Calculator - Free Square Root & Cube Root Calculator Online

A root calculator is a useful tool for calculating various types of roots, including square roots, cube roots, and custom nth roots. Roots are the inverse operation of exponents and are fundamental in mathematics, science, and engineering.

How to Use the Root Calculator

1. Select Operation: Choose between Square Root (√x), Cube Root (∛x), or Custom Root (xⁿ).
2. Enter Number: Enter the number you want to find the root of.
3. Enter Root Degree (for Custom Root): If you selected Custom Root, enter the degree (n) of the root (must be greater than 1).
4. Click Calculate: Get the result instantly with accurate decimal precision.

What is a Square Root?

The square root of a number (√x) is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. The square root is denoted by the radical symbol √.

• √4 = 2 (2 × 2 = 4)
• √9 = 3 (3 × 3 = 9)
• √16 = 4 (4 × 4 = 16)
• √25 = 5 (5 × 5 = 25)

What is a Cube Root?

The cube root of a number (∛x) is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. The cube root is denoted by the radical symbol ∛.

• ∛8 = 2 (2 × 2 × 2 = 8)
• ∛27 = 3 (3 × 3 × 3 = 27)
• ∛64 = 4 (4 × 4 × 4 = 64)
• ∛125 = 5 (5 × 5 × 5 = 125)

What is a Custom (Nth) Root?

A custom or nth root is a generalization of square and cube roots. The nth root of a number is a value that, when raised to the power of n, gives the original number. For example, the 4th root of 16 is 2 because 2⁴ = 16.

• 4th root of 16 = 2 (2⁴ = 16)
• 5th root of 32 = 2 (2⁵ = 32)
• 6th root of 64 = 2 (2⁶ = 64)

Root Calculation Formula

The nth root of a number x can be calculated using the formula:

ⁿ√x = x^(1/n)

For example:

• Square Root: √x = x^(1/2)
• Cube Root: ∛x = x^(1/3)
• 4th Root: ⁴√x = x^(1/4)

Properties of Roots

• Product Rule: √(ab) = √a × √b
• Quotient Rule: √(a/b) = √a / √b
• Power Rule: (ⁿ√x)ᵐ = ⁿ√(xᵐ) = x^(m/n)
• Root of Root: ᵐ√(ⁿ√x) = ᵐⁿ√x

Applications of Roots

• Geometry: Used in calculating distances, areas, and volumes
• Physics: Used in formulas for motion, energy, and waves
• Engineering: Used in structural analysis and design calculations
• Statistics: Used in calculating standard deviation and variance
• Finance: Used in compound interest and investment calculations

Perfect Squares and Cubes

Perfect squares are numbers that have exact integer square roots, while perfect cubes have exact integer cube roots:

• Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
• Perfect Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Negative Numbers and Complex Roots

For real numbers:

• Square Root: The square root of a negative number is not a real number (it's an imaginary number). For example, √(-4) = 2i, where i is the imaginary unit.
• Cube Root: The cube root of a negative number is negative. For example, ∛(-8) = -2.
• Odd Roots: Odd roots (cube, 5th, 7th, etc.) of negative numbers exist and are negative.
• Even Roots: Even roots (square, 4th, 6th, etc.) of negative numbers are not real numbers.