how could we solve the equation
$x^{n}-dy^{n}=1 $
by knowing the continued fraction expansion of $ d^{1/n} $ ??
in case $ n=2 $ is pell's equation
if I divide all by $ y^{n} $ then
$ (\frac{x}{y})^{n}-d= \frac{1}{y^{n}}$
so if $ y $ is big then i have the equality taking the n-th roots
$ \frac{x}{y}=d^{1/n} $
You can try that, but be aware that Thue's Theorem (1909) says that there are only a finite number of integer solutions once your $x^n - d y^n$ is irreducible. See Chapter 22, Diophantine Equations, by Louis J. Mordell, especially page 186. The theorem and proof are very much about continued fractions; the general topic is called Diophantine Approximation, there are entire books on just that.