Square Root & Cube Root Calculator
Calculate square roots, cube roots, and any nth root. Shows the exact result, decimal approximation, and whether the number is a perfect square or cube.
Square Roots Guide
What Is a Square Root?
The square root of a number n is the value x such that x × x = n. Written as √n or n^(1/2). Every positive number has two square roots — positive and negative (√9 = 3 or −3). By convention, √n refers to the positive root. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...) have integer square roots. All other positive numbers have irrational square roots — decimals that never terminate or repeat.
Estimating Square Roots by Hand
To estimate √50: find the nearest perfect squares either side (49 and 64). √49 = 7, √64 = 8. 50 is 1/15 of the way from 49 to 64, so √50 ≈ 7 + 1/15 ≈ 7.07. The Babylonian method (iterative): guess g = 7, improve: g = (7 + 50/7)/2 = 7.07, improve again: g = (7.07 + 50/7.07)/2 = 7.0711. This converges extremely quickly — the actual value is 7.07107...
Cube Roots and nth Roots
Cube root (∛n): the value x such that x³ = n. Unlike square roots, cube roots of negative numbers are real: ∛(−8) = −2. Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Fourth root: x⁴ = n. The nth root of n is n^(1/n). As n increases, n^(1/n) approaches 1 — the 1,000th root of 2 is only 1.000693.
Applications
Square roots appear in: the Pythagorean theorem (c = √(a²+b²)), standard deviation (√variance), physics (escape velocity, pendulum period), finance (volatility calculations in options pricing, RMS voltage), and geometry (diagonal of a square = side × √2 ≈ 1.414). Cube roots appear in: volume calculations (side of a cube = ∛volume), thermodynamics, and in determining the radius from a sphere's volume.
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