Square Root Calculator

The square root of a number x is the value that, when multiplied by itself, gives x. Roots appear constantly in geometry, physics, statistics, and engineering. Understanding both perfect and imperfect square roots — and how to calculate nth roots — expands your mathematical toolkit significantly.

Square root formula: √x = x^(1/2)

nth root: ⁿ√x = x^(1/n)

Perfect squares (exact answers): √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10 √121=11, √144=12, √169=13, √196=14, √225=15, √256=16, √289=17, √324=18, √361=19, √400=20

Worked examples:

Square root: √144 = 12 (because 12 × 12 = 144) Square root (imperfect): √50 = 7.071 (because 7.071² ≈ 50) Cube root: ³√27 = 3 (because 3³ = 27) 4th root: ⁴√256 = 4 (because 4⁴ = 256) 5th root: ⁵√100,000 = 10 (because 10⁵ = 100,000)

Practical applications:

• Geometry: Side of a square = √(Area). A square room with 144 sq ft has sides of √144 = 12 feet.
• Pythagorean theorem: Hypotenuse = √(a² + b²). A right triangle with legs 3 and 4: √(9+16) = √25 = 5.
• Standard deviation: In statistics, σ = √(variance) — the square root appears in nearly every statistical formula.
• Physics: Velocity from kinetic energy: v = √(2KE ÷ m).
• Finance: CAGR involves the nth root: CAGR = (End Value ÷ Start Value)^(1/n) − 1.

Simplifying square roots: √72 = √(36 × 2) = 6√2 ≈ 8.485 — factor out perfect squares for cleaner expressions.