Answer
See the detailed answer below.
Work Step by Step
We know that the electric field is given by
$$E= \dfrac{-dV}{dx} $$
Hence its magnitude is given by
$$E=\left| \dfrac{-dV}{dx}\right|$$
So the electric field dots are given by dividing the given potential differences by the separation distance between the two tips [which is 1 mm].
Thus,
\begin{array}{|c|c|c|c|}
\hline
\Delta V\;{(\rm V)}& E\;({\rm V/m})&r\;({\rm m})\\
\hline
0.0347& 34.7& 0.02 \\
\hline
0.0066& 6.6&0.04 \\
\hline
0.0021 & 2.1& 0.06 \\
\hline
0.0012 & 1.2&0.08 \\
\hline
0.0006& 0.6 &0.10 \\
\hline
\end{array}
And we are given that
$$E=\dfrac{C}{r^n}$$
Taking the normal logarithm for both sides;
$$\ln(E)=\ln \left[\dfrac{C}{r^n}\right]$$
$$\ln(E)=\ln(C)-n\ln(r)$$
This is a straight line formula $y=mx+b$ where $y=\ln(E)$, $x=\ln(r)$, and $m=-n=\rm slope$, and $b=\ln(C)$ which is the the $y$-intercept.
Plug the dots from the table above after adding the algorithm.
The slope of the best-fit line is -2.5, as we see below.
Thus,
$$n=-\rm Slope=-(-2.5)$$
$$n=\color{red}{\bf 2.5}$$
The $y$-intercept of this line is at -6.23.
Thus,
$$\ln(C)=-6.23$$
$$C=\color{red}{\bf 1.97\times 10^{-3}}\;\rm V\cdot m^{3/2}$$
