Changing Stoichiometric Coefficients

[Kyle Reidy 3H]

One of the questions on the practice final asked how the equilibrium constant would change if we halved all the stoichiometric coefficients. The answer is that we would ultimately get the square root of the original constant. This concept is confusing me though.

I believed we could always write some equation, say
A(g) + B(g) --> X(g) + 2Y(s)
equivalently if we doubled all the coefficients, like
2A(g) + 2B(g) --> 2X(g) + 4Y(s)

The ratios in the second equation are just like the first and describe 2 moles of element A reacting with 2 moles of element B. This should theoretically produce twice the amount of product as the reaction with one mole of each reactant, as described in the first equation. However, if we calculate the chemical equilibrium, we would find that
K= [X]²/([A]²*[B]²) is in fact different from K= [X]/([A]*[B]). Is there something I am missing? Are the equations describing two different reactions?

[Michaela Capps 1l]

The coefficients are superscripts for equilibrium expressions so when it said half coefficients, I put the half in the superscript of the x. If it said double all coefficients, you would have x^2

[Kyle Reidy 3H]

Yes, I understand that part, but it's more a conceptual question. I would have thought that as long as the ratio remains the same, it wouldn't matter if you doubled every stoichiometric coefficient or put Avogadro's number in front of every coefficient. Maybe I'm confused about the definition of the constant?