Mean question?

Let the prime factorization of the number be $N = p_1^{e_1}\cdot p_2^{e_2}\cdot p_3^{e_3}\cdot p_4^{e_4}.$ The eighth power of the geometric mean of $\{p_1, p_2, p_3, p_4\}$ is $N = \left(\sqrt[4]{p_1\cdot p_2\cdot p_3\cdot p_4}\right)^8 = p_1^2\cdot p_2^2\cdot p_3^2\cdot p_4^2.$ The smallest number with this structure is, of course, that with the smallest values of $p$ : $N = 2^2\cdot 3^2\cdot 5^2\cdot 7^2 = 210^2 = \boxed{44100}.$