Square Root Calculator

The square root of a number x is the value that, when multiplied by itself, gives x. Roots turn up constantly in geometry, physics, statistics and engineering, usually the moment an area or a squared term needs undoing.

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 is what puts the answer back into the same units as the data.
• Physics: Velocity from kinetic energy: v = √(2KE ÷ m).
• Finance: CAGR (Compound Annual Growth Rate) involves the nth root: CAGR = (End Value ÷ Start Value)^(1/n) − 1.

Simplifying radicals by hand

Factor the number into primes, then group the primes in lots matching the index of the root. Each complete group escapes the radical as a single copy of that prime, and whatever cannot form a complete group stays inside.

√72 is the standard example. 72 = 2³ × 3², which for a square root groups in pairs: the 3² is a complete pair so a 3 comes out, and 2³ gives one complete pair plus a spare 2 that stays behind. That leaves 6√2, about 8.485.

The same procedure works for any index, which is the part most people never get told. ³√54 factors as 2 × 3³, and for a cube root you group in threes, so the 3³ comes out whole and the 2 stays: ³√54 = 3³√2. The calculator prints these steps for whichever root you pick.

A number simplifies to a whole number only when every prime in it appears a multiple of the index times. That is exactly what makes 64 a perfect square (2⁶), a perfect cube (2⁶), and a perfect 6th power all at once.