The minimum value of the sum of real numbers $a^{-5}, a^{-4}, 3 a^{-3}, 1, a^8$ and $a^{10}$ with $a0$ is

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

Aicharya · Accepted Answer

Using $A M \geq G M$, $ \begin{aligned} & \frac{a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+1+a^8+a^{10}}{8} \ & \geq\left(a^{-5} \cdot a^{-4} \cdot a^{-3} \cdot a^{-3} \cdot a^{-3} \cdot 1 \cdot a^8 \cdot a^{10}ight)^{\frac{1}{8}} \ & \Rightarrow a^{-5}+a^{-4}+3 a^{-3}+1+a^8+a^{10} \geq 8 \cdot 1 \end{aligned} $ Hence, minimum value is 8

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

Solution