<s>[INST] Number of real solutions of the equation $1+8^x+27^x = 2^x+12^x+9^x$ 
Find the number of real solutions $x\in\mathbb{R}$ of the equation
  $$
1+8^x+27^x = 2^x+12^x+9^x
$$

My Attempt:
Let $2^x=a>0$ and $3^x=b>0$ where $x\in \mathbb{R}$. This allows us to change the equation to
$$
1+a^3+b^3 = a+a^2b+b^2
$$
This can be rewritten as
$$
(a+b)^3-3ab(a+b)+1 = a+ab(a+b)
$$
How can I solve the problem from this point?
 [/INST] Hint: by rearrangement inequality
$$a^3+b^3+c^3 = a^2b+b^2c+c^2a$$
happens iff $a=b=c$. 
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