How to Calculate Square Root — With and Without a Calculator

The square root of a number x is the value that, multiplied by itself, gives x. For perfect squares like 25 or 144, the answer is a clean whole number. For most other numbers, the square root is irrational, and estimating it by hand takes a bit of technique. This guide covers both the quick estimation method and how to use a calculator correctly for exact results.

Perfect Squares to Memorize

1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 13²=169, 14²=196, 15²=225. Memorizing these makes estimating any square root much faster.

Estimating a Square Root by Hand

To estimate √50: find the two perfect squares it falls between. 49 = 7² and 64 = 8², and 50 is just above 49, so √50 is just above 7 — approximately 7.07 (the actual value). To refine the estimate: since 50 is 1 unit above 49 (out of a 15-unit gap to 64), estimate roughly 7 + (1÷15) ≈ 7.07, which matches the true value closely.

The Long Division Method (Exact, by Hand)

1. 1.Pair the digits of the number from the decimal point, working outward in both directions.
2. 2.Find the largest number whose square is less than or equal to the first pair — this is the first digit of your answer.
3. 3.Subtract, bring down the next pair, and double your current answer to form a new divisor.
4. 4.Find the next digit by trial — the digit that, appended to the divisor and multiplied by itself, is less than or equal to the remainder.
5. 5.Repeat for each pair of digits, including pairs after the decimal point, for more decimal precision.

Using a Calculator Correctly

Most calculators have a dedicated √ button. Enter the number, then press √ (on some calculators you press √ first, then the number). For numbers that are not perfect squares, the calculator gives a long decimal — round to the precision your problem requires (usually 2–4 decimal places for most practical purposes).

Square Roots of Negative Numbers

There is no real number that squares to give a negative result (since any real number squared is positive or zero). The square root of a negative number is called an imaginary number, written using i, where i = √−1. For example, √−9 = 3i. This only matters in advanced algebra and engineering contexts — for everyday calculations, a negative number under a square root usually signals an error.

Cube Roots and Other Roots

The same calculator logic extends to cube roots (∛) and higher roots — the cube root of x is the value that, cubed, gives x (e.g., ∛27 = 3, since 3³=27). Unlike square roots, cube roots of negative numbers ARE real (∛−27 = −3), since a negative number cubed stays negative.