The minimum value of the sum of real numbers `a^(-5),a^(-4),3a^(-3),1,a^(8)" and "a^(10)" with "agt0` is

Explanation

To find the minimum value of the sum \( S = a^{-5} + a^{-4} + 3a^{-3} + 1 + a^{8} + a^{10} \) for \( a > 0 \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Terms**:
The terms in the sum are \( a^{-5}, a^{-4}, a^{-3}, a^{-3}, a^{-3}, 1, a^{8}, a^{10} \).

2. **Count the Number of Terms**:
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