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___

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Solution

Using $A M \geq G M$
$\frac{a^{- 5} + a^{- 4} + a^{- 3} + a^{- 3} + a^{- 3} + 1 + a^{8} + a^{10}}{8} \geq (a^{- 5} . a^{- 4} . a^{- 3} . a^{- 3} . a^{- 3} .1 . a^{8} . a^{10})^{\frac{1}{8}}$
$\Rightarrow a^{- 5} + a^{- 4} + 3 a^{- 3} + 1 + a^{8} + a^{10} \geq 8.1$
Hence, minimum value is 8.

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