Root Calculator

Calculate the cube root of a number. Cube root formula rⁿ=x Cube root calculation √4=2 Calculate the nth root of a number. nth root formula nth root calculation Root index

What Is a Root?

The nth root of a number x is a value r such that rⁿ = x. In other words: what number, multiplied by itself n times, equals x?

Notation: √ⁿ(x) or x^1/n.

Square root (n=2): √x. What number squared gives x?

Cube root (n=3): ³√x. What number cubed gives x?

nth root: √ⁿ(x). For any integer n ≥ 2.

Properties of Roots

• √ⁿ(x x y) = √ⁿ(x) x √ⁿ(y) — root of a product equals product of roots.
• √ⁿ(x/y) = √ⁿ(x) / √ⁿ(y) — root of a quotient equals quotient of roots.
• (√ⁿ(x))^m = √ⁿ(x^m) — powers and roots can be exchanged.
• √ⁿ(√^m(x)) = √^nm(x) — nested roots multiply their indices.

Square Root Simplification

Many square roots simplify by extracting perfect square factors:

√72 = √(36 x 2) = √36 x √2 = 6√2

√50 = √(25 x 2) = √25 x √2 = 5√2

The calculator gives the decimal value. For simplified radical form, factor the radicand and extract perfect powers.

Applications

Geometry: Pythagorean theorem requires square roots. Diagonal of a square = a√2.

Physics: RMS (root mean square) calculations for voltage, current, and velocity.

Statistics: Standard deviation is the square root of variance.

Finance: Compound annual growth rate (CAGR) involves nth roots.

Engineering: Natural frequency calculations, damping ratios, and signal processing all use roots.

Negative Radicands

Even roots (2, 4, 6...) of negative numbers produce complex results (involving i = √(-1)). Odd roots (3, 5, 7...) of negative numbers give real negative results: ³√(-8) = -2 because (-2)³ = -8.