What FORMULA of "nth ROOT" ? How do to obtain it with just unique formula?

https://en.wikipedia.org/wiki/Nth_root_algorithm

Unfortunately, there’s no explicit formula for finding a root — not even for the simplest case of a square root.

Calculators and computers have dedicated circuits and software to iterate through an algorithm like the one on Wikipedia quickly and easily. If you, me, or anyone were able to discover a direct formula to do the same thing it might result in some new branch of mathematics.

I’m curious to see an example problem you’re looking at just to make sure I fully understand your question. Have fun number crunching!

Are you permitted to use logarithms? If yes, then

The nth root of x = 10^(log(x)/n)

@LuckyGuy That’s a good method — it’s how some calculators do it, with log tables and fast exponentiation algos — but if you have a calculating machine, might as well just type in x ^ (1/n) .

@hrairoo I could have used natural logs but I have a better feel for log 10. I keep some of the numbers in my head so I can make an estimate. Yeah… that’s how I roll.

It is convenient to work base 10, because you can often easily estimate the root to within a factor of 10 or less. For example, cube root of 20 million. 20 million = 20×10^^6, so cube root is (cube root of 20) x 10^^2 = (cube root of 20) x 100, which between 200 and 300.

You can also get an estimate of the root of a number to within a factor of 2 by using binary representation. 20 million is of binary order 25 (2^^25). To estimate the cube root, we have 24/3 = 8, so the cube root of 20 million is between 2^^8 (256) and 2^^9 (512). Combined with what we found for base 10, we get a number between 256 and 300. The actual cube root rounds to 271. Not too bad an estimate for minimal effort.