Recently, I have asked how to approximate cube roots to at least one decimal digit. I want to do the same for $n$th roots ($n$ is an integer).
Here's what I have done:
Let $\sqrt[n]{a}$ be the number we are trying to approximate. Let $c$ be the closest $n$th power to $a$. Then, by Newton's method (for the real zero of $x^n-a=0$) we have $$\sqrt[n]{a}\approx c-\frac{c^n-a}{nc^{n-1}}=\frac{(n-1)c^n-a}{nc^{n-1}}$$
The problem with this approximation is that it is bad for big $n$. Substituting this expression back into $c$ is too tedious to do. Any better ideas?
I assume $a > 0$ and $n > 0$. Let $b^n$ be the greatest $n$-th power $\le 10^n a$. Then $$\frac{b}{10} \le a^{1/n} < \frac{b+1}{10}$$ so $b/10$ is an approximation to within one decimal digit.