Question

The minimum value of the sum of real numbers $a^{-5},a^{-4},3a^{-3},1,a^{8}and{a}^{10}$. With a > 0 is___

Answer 1

$a>0,\frac{a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+1+a^8+a^{10}}{8}\ge 1$ (Applying AM GM)
$\Rightarrow a^{-5}+a^{-4}+3a^{-3}+1+a^8+a^{10}\ge 8$
So, the required minimum value is 8