Take the prime factorization of [tex]x[/tex].
[tex]x = 7\times24\times48 = 7\times(2^3\times3)\times(3\times2^4) = 2^7\times3^2\times7[/tex]
If [tex]xy[/tex] is a perfect cube, then the smallest [tex]y[/tex] that makes this happen is [tex]y=2^2\times3\times7^2 = \boxed{588}[/tex]. We "complete" the cube by introducing just enough factors to get each prime power to be a multiple of 3.
