Suppose $t$ is a positive integer such that $\mathop{\text{lcm}}[12,t]^3=(12t)^2$. What is the smallest possible value for $t$?

LCM[12, t]^3 =(12t)^2, solve for t

t = 18, so that:

LCM[12, 18]^3 =LCM[1728, 5832] =46,656

46,656 =(12t)^2=144t^2

t^2 =46,656 /144 =324

t = sqrt(324) = 18