Simplify $\dfrac{8^{x+1} \cdot 2^{x-1}}{4^{x+2}}$.

Simplify $\dfrac{8^{x+1} \cdot 2^{x-1}}{4^{x+2}}$.
  1. $2^{2x+1}$
  2. $2^{2x-2}$
  3. $2^{x+1}$
  4. $4^{x-1}$

Solution

Convert to base 2: $\dfrac{2^{3(x+1)} \cdot 2^{x-1}}{2^{2(x+2)}} = 2^{(3x+3)+(x-1)-(2x+4)} = 2^{2x-2}$.

Asked in: IMO

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