If $2^{m} \cdot 2^{n} = 2^{6}$ and $\dfrac{2^{m}}{2^{n}} = 2^{2}$, find $m$ and $n$.

If $2^{m} \cdot 2^{n} = 2^{6}$ and $\dfrac{2^{m}}{2^{n}} = 2^{2}$, find $m$ and $n$.
  1. $m = 4,\ n = 2$
  2. $m = 2,\ n = 4$
  3. $m = 3,\ n = 3$
  4. $m = 6,\ n = 0$

Solution

From the two equations, $m + n = 6$ and $m - n = 2$. Adding: $2m = 8 \Rightarrow m = 4$, hence $n = 2$.

Asked in: IMO

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