Exponentiation to Addition

The $y$ factor below indicates how many $x$ we should add, as is the nature of multiplication.

$\large \begin{aligned} x^{x} = & \space x+x+x+...+x \\ x^{x} = & \space xy \\ \frac{x^{x}}{x} = & \space y \\ \boxed{x^{x-1}} = & \space y \end{aligned}$

$x^x = x + x + x + x + ...$

Assume $\boxed {x = 4}$ (for easy explanation)

$4^4 = 4 + 4 + 4 + 4 + ...$

$4^4 = 4^1 + 4^1 + 4^1 + 4^1 + 4^1 + ...$

$4^4 = 4^2 + 4^2 + 4^2 + 4^2 + 4^2 + ...$

$4^4 = 4^3 + 4^3 + 4^3 + ...$

$4^4 = 64 + 64 + 64 + 64$

$4^4 = 64 × 4$

We need to add $x$ $\boxed {x^{x-1}}$ times to make $x^x$ , which means:

$\boxed {x^1 × x^{x-1} = x^x}$

$\boxed {4^1 × 4^{4-1} = 4^4}$