After some shallow research, I've found no results of anyone asking the same question as me. Please feel free to refer me to wherever this has been discussed previously.
When expressing a change with the letter delta: $\Delta x$ for example and raise it, let's say, to the third power, should it be written as $(\Delta x)^3$, $\Delta^3 x$, or $\Delta x^3$?
I'm guessing the last one is equivalent to $\Delta(x^3)$.
Thanks in advance.
Write it as $(\Delta x)^3$.
The others would probably be interpreted like so:
$\Delta^3 x$ looks like a higher-order finite difference. This is the finite analogue of $\dfrac{\rm d^3}{{\rm d}x^3}x$. The MathWorld link shows an example with $\Delta^3$ applied to the sequence of cubes $1, 8, 27, \ldots$.
$\Delta x^3$ looks like the change in $x^3$, not the cube of the change in $x$.
Incidentally, note how in $\frac{\rm d^3}{{\rm d}x^3}x$, there are no parentheses. The denominator is really $({\rm d}x)^3$, but for historical reasons we don't write it that way.