Answer
$$\frac{c}{n}$$
Work Step by Step
We find the derivative:
$$
f^{\prime}(x)=nx^{n-1}
$$
Then at $x=c, m=f^{\prime}(c)=nc^{n-1}$. Hence, the tangent line is:
$$
\begin{aligned}
\frac{y-y_{1}}{x-x_{1}} &=m \\
\frac{y-c^n}{x-c} &=nc^{n-1} \\
y &=nc^{n-1} (x-c)+c^n
\end{aligned}
$$
Since the tangent line intersects with the $x-$axis at $x=0,$ then $Q$ has the coordinates $(c-\frac{c}{n},0), R$ has coordinates $(c,0)$, and the subtangent is
$$
c-\left(c-\frac{c}{n}\right)=\frac{c}{n}
$$
