Decades ago, when the slide rule was the norm, and calculators with the ability to calculate square roots were the cutting edge of technology, I was given an interesting formula. It went like this:
To estimate the $R$th root of $n$ using a simple calculator (where both $R$ and $n$ are positive numbers; fractions are allowed), do the following.
- Enter $n$ into the calculator.
- Press the $\sqrt{}$ button 12 times (i.e. calculate the 4096th root of $n$).
- Subtract 1.
- Divide by $R$.
- Add 1.
- Square the result (usually press the $\times =$ buttons) 12 times (i.e. calculate the 4096th power of the result).
Fascinatingly, this works pretty well.
For example, using this method on my calculator to find $\sqrt[8.5]{586.426}$ gives the answer with an error of just 0.05%.
This means that: $$\sqrt[R]{n} \approx \biggl(\frac{\sqrt[4096]{n} - 1}{R} + 1\biggr)^{4096}$$
Now, to me, the number $2^{12}$ seems arbitrary. So, my question is:
Why does this work?
I've tried muddling through equations to no avail.
EDIT: I've just spotted this question, which seems relevant.