$a^2 = 2b^3 = 3c^5$ Find the smallest value of $abc$. We have  following equation:
$a^2 = 2b^3 = 3c^5$
Where $a, b, c$ are natural numbers.
Find the smallest possible value  of product $abc$.

Since $a^2=2b^3$, this means $a = \sqrt{2b} \cdot b$. Hence, $b=2l^2$. This gives us $a=4l^3$.
We also need $a^2 = 3c^5 \implies 16l^6 = 3c^5 \implies c = \left(\dfrac{2^4l}3\right)^{1/5} \cdot l$. This means $l = 3\cdot 2 \cdot m^5$.
Hence, we have $(a,b,c) = (864m^{15},72m^{10},12m^6)$. The minimum is obtained when $m=1$. Hence, $(a,b,c) = (864,72,12)$.
