I have to two expressions: (||x-ay||^4)/a^3 and (||x-ay||^3)/a^2, where the real vector x and the nonnegative scalar a are the optimized variables. The real vector y is known.

These two expressions are joint convex in (x, a). How can I express them in the acceptable forms?

For the first one, add an extra a as an argument of geo_mean in the above link. Similarly, you’ll need to restrict a > 0, which the geo_mean` formulation automatically does.