Suppose that $a$ and $b$ are positive integers with $2^{a} \times 3^{b}=324$.

Evaluate $2^{b} \times 3^{a}$.
Repeatedly dividing, $324=3 \times 108=3^{2} \times 36=3^{2} \times 3^{2} \times 2^{2}=2^{2} \times 3^{4}$.

Therefore, $a=2$ and $b=4$, so $2^{b} \times 3^{a}=2^{4} \times 3^{2}=16 \times 9=144$.