If we let $z=xy$, we have: $$z=\sum_{i=1}^{y}x$$ So multiplication can be written as iterative sum. Likewise, if we have $z=x^y$, then we can write: $$z=\prod_{i=1}^{y}x$$ But how to write the exponent in sum form instead of product form?
I tried to experiment by writing for example $5^2=(5+5+5+5+5), 2^4=((2+2)+(2+2))+((2+2)+(2+2))$, but I came up with nothing.
Edit: From the conversation in the comments, you can write exponent as nested summations:
$$z=\sum_{i=1}^{x}\sum_{j=1}^{x}\sum_{k=1}^{x}\dots\text{(y times)}\dots x$$
Does this work?:
Let $z = x^y$. This will work for $x,y \in \mathbb{N}$.
$$ z = x^y \\ = \sum\limits_{i_1 = 1}^{x} x^{y-1} \\= \sum\limits_{i_1 = 1}^{x} \left( \sum\limits_{i_2 = 1}^{x} x^{y-2} \right) \\ = \sum\limits_{i_1 = 1}^{x} \left( \sum\limits_{i_2 = 1}^{x} \left( \sum\limits_{i_3 = 1}^{x} x^{y-3} \right) \right) \\ = \underbrace{\sum\limits_{i_1 = 1}^{x} \left( \sum\limits_{i_2 = 1}^{x} \left( \sum\limits_{i_3 = 1}^{x} \left( \cdots \left( \sum\limits_{i_y = 1}^{x} 1 \right) \right) \right) \right)}_{y \text{ times}} $$