Square Root Calculator

Formula

The nth root of a number x is equivalent to raising x to the power of 1/n. The square root is x^(1/2), the cube root is x^(1/3), and so on. This is why square root and exponentiation are inverse operations.

How to use the Square Root Calculator

1. 1

   Enter your number

2. 2

   Enter your root (n)

   2 for square root, 3 for cube root, etc.

3. 3

   Read your results instantly

   Results update in real time as you type.

Perfect squares and irrational numbers

A perfect square is any integer that is the square of another integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... The square root of these numbers is always a whole number.

For all other positive integers, the square root is an irrational number — a decimal that never terminates or repeats. √2 ≈ 1.41421356... and goes on forever. This was a significant discovery in ancient Greece; legend says Hippasus was drowned for proving it.

Square roots in geometry

Square roots appear everywhere in geometry. The Pythagorean theorem (a² + b² = c²) requires taking a square root to find the hypotenuse. The diagonal of a unit square is √2. The area of a circle uses the square root in reverse: r = √(A/π).

In physics, the period of a pendulum is proportional to the square root of its length. In statistics, standard deviation is the square root of variance. Square roots show up wherever you're converting between squared quantities and linear ones.

Estimating square roots mentally

You can estimate square roots by finding the two perfect squares between which your number falls. √50: 7² = 49 and 8² = 64, so √50 is just over 7. More precisely, 50 is 1/15 of the way from 49 to 64, so √50 ≈ 7.07.

Another approach: the Babylonian method (also called Newton's method for square roots) iterates: guess g, then refine as (g + x/g)/2. Starting with g=7 for √50: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071. One step gives four decimal places of accuracy.

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Worked Examples

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