Its a numerical analysis question where $\Delta$ is the forward difference operator. But although I have learned about the different operators I really can't understand the meaning of $0^m$. The answer options are
(a) 500 (b)515 (c)530 (d)540
I am not able to proceed because of the $0$ which I am not able to understand properly.
I don't understand the $0^m$ either, but we can still run through the thing symbolically. We know that $$\Delta^30^6 = \Delta^3 x^6\rvert_{x=0},$$ and this in turn is \begin{align*} \Delta^3 x^6 &=\Delta^2 [(x+1)^6 - x^6]\\ &=\Delta [((x+2)^6 - (x+1)^6) - ((x+1)^6 - x^6)]\\ &=[((x+3)^6 - (x+2)^6) - ((x+2)^6 - (x+1)^6)]\\&\qquad-[((x+2)^6 - (x+1)^6) - ((x+1)^6 - x^6)]\end{align*}evaluated at $x=0$ to get 540.
Still no clue what $0^6$ means in this context. Would be interested if anyone knows.