Linear approximation of $\sqrt[7]{e}$ I need to find a linear approximation $\sqrt[7]{e}$. I know that for $x_0 = a + \Delta x$
$$
f(x_0) \approx f(a)+f'(a)\cdot \Delta x
$$
Thus my inital function is $f(x) = \sqrt[7]{x}$ and $e = 1 + (e - 1)$, so
$$
f(e) \approx \sqrt[7]{1} + \frac{1}{7}1^{\frac{-6}{7}}\cdot(e - 1) = \frac{6}{7} + \frac{e}{7} \approx 1.2455.
$$
However, I can't say that I'm entirely pleased by the result. Even though one could argue that $f(x) = \sqrt[7]{x}$ grows in such slow manner that approximating $f(e)$ by $f(1)$ is good enough for linear approximation, I'm not convinced. So I started to wonder: does there exist a function $g(x): f(g(x)) = \sqrt[7]{g(x)}$ and $f(g(c)) = \sqrt[7]{g(c)}= \sqrt[7]{e}$ that could provide a more precise approximation?

$[k,k]$ Padé approximants $P_k$ of $e^{\frac{1}{n}}$ are not bad
$$P_1=\frac{2 n+1}{2 n-1}\qquad P_2=\frac{12 n^2+6 n+1}{12 n^2-6 n+1} \qquad P_3=\frac{120 n^3+60 n^2+12 n+1}{120 n^3-60 n^2+12 n-1}$$ and so on are more then decent. For $n=7$ they generate
$$\left(
\begin{array}{ccc}
k & P_k & \log_{10} \left(\left| P_k-e^{\frac{1}{7}}\right| \right)\\
 1 & \frac{15}{13} & -3.55 \\
 2 & \frac{631}{547} & -7.02 \\
 3 & \frac{44185}{38303} & -10.86 \\
 4 & \frac{4330761}{3754241} & -14.95 \\
 5 & \frac{545720071}{473072669} & -19.24 \\
 6 & \frac{84045221695}{72856945267} & -23.68 \\
\end{array}
\right)$$
