Why can't we use the rules for exponents when the bases are not common?

Ok, the rules of the exponent come from a logic construction.

If we have x^n

this means that n is multiplied by itself n times.

If we decompose n into a + b, we have:

x by itself a times, and then x by itself b times, and for how the product works, this is equivalent:

if n = 5, a = 2 and b = 3

x^5 = (x*x*x*x*x) 5 times-

x^5 = x^ (2 + 3) = (x^2) * (x^3) = (x*x*) * (x*x*x) = x*x*x*x*x = x^5

And the same for the other rules:

(x^n) ^b = x^n*b and such.

Obviusly, this only works when we have a common base.

So the correct answer is that we constructed the exponential rules in a way that only can be used when we have a common base, and this happens because to construct them, we started with common bases.