More manipulating equilibrium constants

If two reactions are summed to give a third reaction. The equilibrium constant for the third reaction is the product of the equilibrium constants for the summed reactions.

Note how the species which cancel when the equations are summed also cancel when the reaction quotient expressions are multiplied.
AgCl(s) Ag+(aq) + Cl(aq)

K = K1 = [Ag+][Cl]

Ag+(aq) + 2NH3(aq) Ag(NH3)2+(aq)
 
K = K2 =  [Ag(NH3)2+]
[Ag+][NH3]2

AgCl(s) + 2NH3(aq) Ag(NH3)2+(aq) + Cl(aq)
 
K3 =  K1 × K2 =  [Ag+][Cl] ×  [Ag(NH3)2+]
[Ag+][NH3]2