The minimum value of the sum of real numbers `a^-5, a^-4, 3a^-3, 1,a^8` and `a^10` with `agt 0` is __________.

Explanation

`:. AMge GM`
`:.(a^(-5)+a^(-4)+a^(-3)+a^(-3)+a^(-3)+1+a^(8)+a^(10))/(8) ge (a^(-5)*a^(-4)*a^(-3)*a^(-3)*a^(-3)*1*a^(8)*a^(10))^((1)/(8))=(1)^((1)/(8))=1`
`implies (a^(-5)+a^(-4)+3a^(-3)+1+a^(8)+a^(10))/(8) ge 1`
`:. a^(-5)+a^(-4)+3a^(-3)+1+a^(8)+a^(10) ge 8`
`implies " Required minimum value " = 8`.