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AoPS Wiki talk:Problem of the Day/September 8, 2011

Solution

We see complicated expressions, so we replace them with variables. Let $a=2^x-4$ and $b=4^x-2$. Then our equation reads $a^3+b^3=(a+b)^3$. Expanding the left gives: \[a^3+b^3=a^3+b^3+3ab(a+b)\] which in turn gives $ab(a+b)=0$. This means that either $a=0$, $b=0$, or $a+b=0$.

If $a=0$, then $2^x-4=0$ and $\boxed{x=2}$.

If $b=0$, then $4^x-2=0$ and $\boxed{x=\frac{1}{2}}$.

If $a+b=0$, then $4^x+2^x-6=0$ and $\boxed{x=1}$.

The sum is $2+\frac{1}{2}+1=\boxed{\frac{7}{2}}$.

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