Simplifying the given expressions we proceed as follows:
(5sqrt3)^x
=5^x*(3^1/2)^x
=5^x*3^x/2
=5^x3^u
where u=x/2
(1/2)^(x-3)
=1/2^(x-3)
=2^-(3-x)
=2^u
where u=-(3-x)
9/3sqrt(3)
=3/(3)^(1/2)
=3(3)^(-1/2)
16/(3sqrt (2^x))
=1/3*(2^4*2^(-x/2))
=1/3*2^(4-x/2)
=1/3*2^u
where:
u=4-x/2
So there are 8 pints in a us gallon, 2 drops of blue to make a pint of purple (2x8x50=800) and then 3 drops of red to make purple (3x8x50=1200) I think maybe I've missed a step but (800+1200=2000) so 2000 drops of red and blue combined to make 50 gallons of purple paint.