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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?

Alex M.
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Han Solo
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    You are assuming that ceiling(x+y)=ceiling(x)+ceiling(y), which is not true in general. In your example, this failure can be seen when x=2 – Alex G. Sep 23 '15 at 12:09

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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.

Suganth Kannan
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