Square Root Calculator

How Does the Formula Work?

The square root of a number n is the value that, when multiplied by itself, gives n. This calculator extends beyond square roots to handle cube roots and any nth root, and it does more than compute decimal approximations — it simplifies radicals by extracting perfect factors using prime factorization.

How Radical Simplification Works

To simplify a radical, the calculator first finds the prime factorization of the number. For a square root, it looks for pairs of prime factors — each pair can be "extracted" outside the radical. For √72: 72 = 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, so 2 × 3 = 6 comes outside, and one 2 remains inside: 6√2. For cube roots, it looks for triples of factors, and for nth roots, groups of n. This process always produces the simplest possible radical expression.

Perfect vs Irrational Roots

A perfect root is an integer — √144 = 12 exactly. Most roots are irrational, meaning they cannot be expressed as a fraction and their decimal expansion continues forever without repeating. The most famous irrational root is √2 ≈ 1.41421356…, which was the first number proven to be irrational by the ancient Greeks. The calculator marks each result as "Perfect" or "Irrational" so you immediately know the nature of the number.

Negative Numbers and Even/Odd Roots

Odd roots of negative numbers are valid real numbers: ∛(−27) = −3 because (−3)³ = −27. Even roots of negative numbers have no real solution — √(−4) does not exist in the real numbers because no real number squared gives a negative result. In the complex number system, √(−4) = 2i, but this calculator works with real numbers only. When you try an even root of a negative number, the calculator displays an appropriate error message.

Applications of Roots

Square roots appear throughout mathematics, science, and engineering. The Pythagorean theorem uses square roots to find distances: the hypotenuse of a right triangle with legs a and b is √(a² + b²). Standard deviation in statistics involves a square root. The quadratic formula contains √(b² − 4ac). In physics, the period of a pendulum involves √(L/g). Cube roots appear in volume calculations — the side length of a cube with volume V is ∛V. Higher roots appear in financial compounding, signal processing, and computer graphics. Radical simplification is a core algebra skill tested at every level from middle school through college.