Let $\mathbb { N }$ denote the set of natural numbers, and let $\left( a _ { i } , b _ { i } \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb { N } \times \mathbb { N }$. Show that there are three distinct elements in the set $\left\{ 2 ^ { a _ { i } } 3 ^ { b _ { i } } : 1 \leq i \leq 9 \right\}$ whose product is a perfect cube.