Suppose I have $f(x) = 5^{\lceil \frac x 3 \rceil}$, where $x \in \Bbb N$.
If I were to simplify $f(x+4)$, can I do the following: $f(x+4) = 5 ^{\lceil \frac {x+4} 3 \rceil} = 5^{\lceil {\frac x 3} \rceil} \cdot 5^{\lceil \frac 4 3 \rceil}$, by the exponent law 1.
Or is this not applicable with the ceiling function?
First, I urge you to graph ceiling function on a calculator to get a good grasp. $$x=1$$ is one counterexample. There are infinitely many. So, the exponent rule does not apply to the ceiling function.