I tried several ways, but i could not come up with any way to have an equation as such:
|n| = ...
without using the absolute value signs on the right side of the equation.
I do not know if there is any way...
I tried using some form of n * i^(an + b) but those efforts were futile.
And I couldn't think of any other way to approach this.
One way is to note that the square root has "branches":
$$(\sqrt n)^2=n\tag 1$$
$$\sqrt{n^2}=|n|\tag 2$$
Note that $(1)$ would dip into the imaginary numbers for a moment if $n\lt 0$. $(2)$ is true because the square root of a number is one of two solutions to the equation involving the square.
$$x^2=4,x=2,-2$$
$$\sqrt 4=2,-\sqrt 4=-2$$