How can you figure out the square root of a number quickly and to a decent degree of accuracy. You can try and figure out which squares the number falls between and try to guess where it falls but how can you do that accurately? Some of you will have heard of a Taylor series. For those who haven’t or for those who wouldn’t mind a refresher a Taylor expansion is a way to approximate a function around a particular value.

For this technique we will use the nearest perfect square number as that value, for example if we were trying to square root 150 we would use 144. Below is the derivation for how we can solve the square root of a number.

Let’s go back to our example the square root of 150. The nearest perfect square is 144 and the difference between this and 144 is 12. By our formula then the square root should be around 12+6/2*12 or 12.25 when the actual answer is 12.247. This method also works for cube roots, the methods is the same and is below.

To take our example again, the nearest cube is 125 and the difference is 25 so our guess for the cube root is 5+25/(3*25) = 5.33. The actual cube root is 5.31. This method is a likely more difficult for most people as cubes are generally not known as well as squares.

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