Square Root Calculator

What a square root is

The square root of a number x is the value that, multiplied by itself, gives x. √49 = 7 because 7 × 7 = 49. Every positive number has two square roots, one positive and one negative (7 and −7), and √ normally means the positive one. Negative numbers have no real square root; their roots are imaginary numbers, such as √−9 = 3i.

Simplifying and estimating

Non-perfect squares can often be simplified by pulling out square factors: √50 = √(25 × 2) = 5√2 ≈ 7.071. To estimate by hand, find the nearest perfect squares: 50 lies between 49 and 64, so √50 is just above 7.

Newton's method refines a guess g with the update (g + x ÷ g) ÷ 2. Starting from g = 7 for x = 50 gives (7 + 50 ÷ 7) ÷ 2 ≈ 7.0714 after one step, already correct to three decimal places. Each further step roughly doubles the number of correct digits.

Where square roots appear

Square roots appear in the Pythagorean theorem, in standard deviation, in the side length of a square of known area (a 2,500 sq ft square plot has 50 ft sides), and in physics formulas such as the period of a pendulum. The square root of a fraction is the root of the top over the root of the bottom: √(9/16) = 3/4.