how do you find Y in Y^10 by hand

Question

I do not want to use a calculator. HOW does is the answer, the answer. HOW does it work.

Answer 1

You can tenth-root the Y^10.

For example you have 1024.
1024 = 2^10, and when you 10th root it, you get 2.

However, that is really hard to do so, as you need an insanely good number sense to do so. What is more, the Y must be a whole number. If it is not, no matter how good you are at numbers, you won't be able to find it.

Another method is to use approximation methods such as newton-rahpson method.
Suppose you know Y^10 = a, where a is some real number.
Then you can let f(x) = Y^10 - a, and work from here to find a good approximation.

Comments

although I actually am "insanely good with numbers" and I find that square rooting thing pretty easy (when the answer is not a decimal), I need to get as accurate as being no more than one-thousandth off. I need this because I am currently trying to create a formula for finding a number with a decimal for an exponent. The equation is      Y^10=B^(10*D)     where Y is the answer, B is the base (the number that is being raised to the power of the exponent), and D is the exponent that is a decimal. Unfortunately, I not only need the solution to Y^10 to be more specific, but also almost all of the answers would be a decimal (or a number with a decimal taged on).