Square Root Calculator

A root is the inverse operation of exponentiation. The square root of a number x is the value that, when multiplied by itself, gives x. Symbolically, √x = y means y × y = x. The square root of 49 is 7 because 7 × 7 = 49. Similarly, the cube root of 27 is 3 because 3 × 3 × 3 = 27. Roots appear in geometry (finding side lengths from areas), physics (calculating speeds and energies), and finance (volatility calculations).

Square roots of negative numbers are not real numbers — they belong to the complex number system. This calculator returns no result for negative inputs when computing even roots (square root, fourth root, etc.). Odd roots like cube roots do accept negative inputs: ∛(−8) = −2 because (−2)³ = −8. The nth root of x is written as ⁿ√x or x^(1/n), and the same parity rules apply.

Perfect squares have exact integer roots: √1 = 1, √4 = 2, √9 = 3, and so on. Most numbers are not perfect squares — √2 ≈ 1.4142, √3 ≈ 1.7321, √5 ≈ 2.2361. These irrational numbers have decimal expansions that never terminate or repeat. Calculators use efficient algorithms like Newton's method to approximate them to many decimal places quickly.

Cube roots (∛) and higher nth roots extend the concept to three or more dimensions. In three-dimensional geometry, if a cube has volume V, its edge length is ∛V. The fourth root appears in some physics formulas. The general formula is: ⁿ√x = x^(1/n). For example, the 4th root of 16 is 16^(1/4) = 2 because 2⁴ = 16.

Key properties simplify calculations: √(a × b) = √a × √b, so √50 = √(25 × 2) = 5√2 ≈ 7.071. √(a/b) = √a / √b. These rules help simplify radical expressions by hand. In practical applications, roots connect to the Pythagorean theorem (c = √(a² + b²)), standard deviation (which involves a square root), and quadratic formula solutions. Select the root type and enter your value to get an instant result.