Root Calculator

Calculate the square root (√), cube root, or any nᵗʰ root. n = 2 for square root, n = 3 for cube root, etc.

Root Calculator — Guide

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Types of Roots and When to Use Them

The square root (√) is the most common, used in geometry (Pythagorean theorem), physics (RMS values), and statistics (standard deviation). The cube root (∛) appears in volume problems — if a cube has volume 125 cm³, its side length is ∛125 = 5 cm. The nth root generalises this: the 4th root of 16 is 2 because 2⁴ = 16. Roots are the inverse of powers, so understanding one means understanding the other.

In finance, the compound annual growth rate (CAGR) formula uses the nth root: CAGR = (End Value / Start Value)^(1/n) − 1, where n is the number of years. A $1,000 investment growing to $1,610 over 5 years has a CAGR of ⁵√(1.61) − 1 ≈ 10% per year.

Root Reference Table

Square, Cube, and Nth Roots

The square root (√x) asks: what number multiplied by itself equals x? √25 = 5 because 5² = 25. The cube root (∛x) asks: what number cubed equals x? ∛27 = 3 because 3³ = 27. The nth root (ⁿ√x) = x^(1/n). Roots and exponents are inverse operations: √(x²) = |x| and (√x)² = x (for x ≥ 0). For non-perfect squares, roots are irrational: √2 ≈ 1.41421356…, a non-terminating, non-repeating decimal.

Roots appear in the quadratic formula (x = (−b ± √(b²−4ac)) / 2a), the distance formula (d = √((x₂−x₁)² + (y₂−y₁)²)), standard deviation (√variance), and RMS voltage. In geometry, the diagonal of a unit square is √2, which was the first irrational number discovered by the ancient Greeks. Computing nth roots efficiently uses Newton's method: xₙ₊₁ = ((n−1)xₙ + A/xₙⁿ⁻¹) / n, which converges rapidly.

Common Root Values

Types of Roots and Their Mathematical Properties

The root of a number is the inverse operation of exponentiation. The square root of x is the number that, when multiplied by itself, equals x. The cube root of x is the number that, when cubed, equals x. In general, the nth root of x is the number raised to the power n that equals x, written as x to the power 1/n. Square roots of positive real numbers always yield two solutions: a positive and a negative value (for example, the square root of 25 is both positive 5 and negative 5). By convention, when asked for "the square root," the positive (principal) root is returned. Cube roots and all odd roots can be negative if the radicand is negative. Even roots of negative numbers are not real numbers but complex numbers. Roots appear throughout geometry (the Pythagorean theorem), physics (velocity, energy), statistics (standard deviation), and engineering (signal amplitude). Irrational roots such as the square root of 2 (approximately 1.41421) cannot be expressed as exact fractions, which is a fundamental result in number theory first proved by ancient Greek mathematicians.

Common Roots Reference Table