multiplied reaction

[Ami_Pant_4G]

If a reaction is multiplied, for example 2N2(g) + 6H2(g) <-> 4NH3(g) from N2(g) + 3H2(g) <-> 2NH3(g), why would the K value be raised to the second power? If the coefficients of the balanced equation can be reduced by some factor why would you need to raise K to some power? Thanks in advance.

[805312064]

If you find K for the original problem, you get something along the lines of K= [NH3}^2 / [N2][H2]^3
The K value for the other problem is K = [NH3}^4 / [N2]^2[H2]^6 which can be simplified to {[NH3}^2 / [N2][H2]^3}^2
Therefore, the K value of the other problem is K^2 of the original problem. I hope this helps!

[Ryan Yee 1J]

Since we take the products over reactants raised to their stoichiometric powers, a multiple of the entire equation would cause for an increase in the same number power for each reactant and product. So if we were to multiply that equation's coefficients by 3 then the resulting K would be K^3

[DanielTalebzadehShoushtari2A]

The above explanations make sense mathematically (thank you for posting them) but I would like to ask:
If each K value is valid for its respective reaction (multiplied or unmultiplied), how can there be 2 different valid K values for what is essentially the same reaction?