1. ((ab^1/3) / (ba^3/2)) ^3/2 = (a^3/2b^1/2) / (b^3/2a^9/4)
2) ((a^-1b^-3/8) / (a^-1/2))^2 = (a^-2b^-3/4) / a^-1
3) (a^3/2b^1/2) / (b^3/2a^9/4) + (a^-2b^-3/4) / a^-1 = (a^3/2b^1/2)*(a^-1) / (b^3/2a^9/4) + (a^-2b^-3/4)(b^3/2a^9/4) / (a^-1) = ((a^1/2b^1/2) + (a^1/4b^3/4)) / (a^-1)(b^3/2a^9/4) = (a^1/2b^1/2 + a^1/4b^3/4) / (a^-1)(b^3/2a^9/4)
4) (1 / (a^-1/4)) + (1 / (b^-1/4)) = (b^-1/4 + a^-1/4) / (a^-1/4b^-1/4)
5) (a^1/2b^1/2 + a^1/4b^3/4) / (a^-1)(b^3/2a^9/4) * (a^-1/4b^-1/4) / (b^-1/4 + a^-1/4) = (a^1/4b^1/4 + b^1/2) / (b^3/2a^5/4)(b^-1/4a^-1/4) = (a^1/4b^1/4 + b^1/2) / (b^5/4a) = b^1/4 (a^1/4 + b^1/4) / b^5/4a = (a^1/4 + b^1/4) / ab