solve: (a^2y^-18=18)/(a^2y^-15=15)

I am not familiar with your notation.

Are you saying

(a^2 y^-18)/(a^2 y^-15) = 18/15 ?

if so, then
y^-3 = 6/5
y^3 = 5/6
y = (5/6)^(1/3)

sub back into
a^2 y^-15 = 15
a^2 (5/6)^-5 = 15
a^2 (6/5)^5 = 15
a^2 = 15(5^5)/(6^5)
a = ± √(15(5^5)/(6^5))