When is this product of powers an integer?

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Let \(r\) and \(s\) be integers. Then \[\frac{6^{r+s}\times12^{r-s}}{8^r\times9^{r+2s}}\] is an integer if

1. \(r+s\le 0\),

2. \(s \le 0\),

3. \(r\le 0\),

4. \(r\ge s\).

Maybe we could break \(6, 12, 8\) and \(9\) down into their prime factors to rewrite the expression?