Square Root Calculator

A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematical Definition: If x² = a, then x is a square root of a Notation: √a = x means x² = a Fundamental Properties: 1. Every positive number has two square roots: ±√a • Principal square root: √a (positive) • Negative square root: -√a Example: √16 = 4 and -4 (because both 4² = 16 and (-4)² = 16) 2. Square root of zero: √0 = 0 3. Square root of negative numbers: Not defined in real numbers • Requires complex numbers: √(-1) = i (imaginary unit) 4. The radical symbol √ refers to the principal (positive) square root Types of Numbers: • Perfect Squares: Numbers with integer square roots Examples: √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6 • Non-Perfect Squares: Numbers with irrational square roots Examples: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 Key Properties: 1. Product Property: √(a × b) = √a × √b 2. Quotient Property: √(a / b) = √a / √b (b ≠ 0) 3. Power Property: (√a)² = a 4. √a² = |a| (absolute value of a) Simplifying Radicals: • Find the largest perfect square factor • Example: √72 = √(36 × 2) = √36 × √2 = 6√2 Applications: Pythagorean theorem, distance formula, quadratic equations, area and volume calculations, physics formulas (kinetic energy, pendulum period), statistics (standard deviation), and engineering calculations.