Why is area expressed as unit^2?

Question

It is clear to me that area is measured in no of unit squares that are enough to cover the given shape's entire surface. So we should say that area of my roof is 50 (meter squares). But why does it equivalently translate into 50 m^2, is it simply an algebraic convenience so that area/length cancellation gives length, and power of 2 also represents the second dimension while unit^3 represents volume because 3 is third dimension?

Answer 1

This is because an area is measured by computing the product of two lengths. For example if those are defined in meters, you multiply some meters by some other meters. Therefore the way to say that the area is in square meters.

The same for volume: you multiply three lengths.

Answer 2

It's just notational convenience. We agreed that the basic unit for measurement of surfaces is the area of a square with side length of $1$ meter and we compare all surface areas we want to measure to that (exactly the same way we simply agreed, that the basic unit for measuring distance is equal to $1$ meter).
We write $m^2$ for this concept because this way we can almost think of the $m$ as a variable and do the usual algebraic manipulations to it.