Review question

# When is this product of powers an integer? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6983

## Solution

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. $r\ge s$.

We have $6 = 2 \times 3, 12 = 2^2 \times 3, 8 = 2^3$, and $9 = 3^2$.

If we separate the powers of $2$ and $3$, we have $\frac{6^{r+s}\times12^{r-s}}{8^r\times9^{r+2s}}= \frac{(2\times3)^{r+s}\times(2^2\times3)^{r-s}}{(2^3)^r\times(3^2)^{r+2s}}$ $= 2^{r+s+2r-2s-3r} \times3^{r+s+r-s-2r-4s}$ $=2^{-s}\times 3^{-4s}.$

This is an integer precisely when $s\le 0$.