The minimum value of the sum of real numbers $a^{-5}, a^{-4}, 3 a^{-3}, 1, a^{8}$ and $a^{10}$ with

Question

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

Professor PC Thomas Classes · Answer

(8) : Using A.M. - G.M. between the numbers, we get\ $\frac{a^{-5}+a^{-4}+3 a^{-3}+1+a^{8}+a^{10}}{8} \geq \sqrt{1}$\ $a^{-5}+a^{-4}+3 a^{-3}+1+a^{8}+a^{10} \geq 8$\ Here the least value is 8 and its realized also at $a=1$.\