Calculate the square root of positive and negative numbers (√x). Provides radical simplification, perfect square check, and imaginary roots.
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A square root of a number x is a number y such that y² = x. Every positive real number has two square roots (one positive and one negative); the non-negative one is called the principal square root, denoted √x.
y = \sqrt{x} \iff y^2 = x \quad (y \ge 0), \quad \sqrt{-x} = i\sqrt{x}The radical symbol denotes the principal non-negative square root. For negative numbers, i represents the imaginary unit √(-1).
Scenario: Find square root of 72
Given: x = 72
Decimal root: √72 ≈ 8.485281
Factor into primes: 72 = 36 × 2
Simplify radical: √(36 × 2) = √36 × √2 = 6√2
Not within real numbers, but in complex numbers: √(-a) = i√a, where i = √(-1). For example, √(-25) = 5i.
By mathematical convention, the radical sign √ denotes the principal (non-negative) root. The equation x² = 25 has two solutions (±5), but √25 = 5.
If a whole number is not a perfect square, its square root is an irrational number with non-repeating, non-terminating decimals (like √2 ≈ 1.4142).
Find the two closest perfect squares. For √50: 7² = 49 and 8² = 64, so √50 is slightly greater than 7 (≈ 7.07).
Calculate the square of any positive or negative number (x² = x × x) with scientific notation and step-by-step arithmetic.
Calculate the principal real cube root of positive and negative numbers. Identifies perfect cubes, nearest integer roots, and complex conjugate roots.
Calculate the nth root of any number (ⁿ√x). Supports even and odd roots, real roots, radical expressions, and step-by-step solutions.