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

Using $\text{A.M.}\ge \text{G.M.}$
$\frac{a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+a^8+a^{10}+1}{8}\ge 1$
$\Rightarrow a^{-5}+a^{-4}+3a^{-3}+a^8+a^{10}+1\ge 8$