  We study the distribution of the generalized gcd and lcm functions on
average. The generalized gcd function, denoted by $(m,n)_b$, is the largest
$b$-th power divisor common to $m$ and $n$. Likewise, the generalized lcm
function, denoted by $[m,n]_b$, is the smallest $b$-th power multiple common to
$m$ and $n$. We derive asymptotic formulas for the average order of the
arithmetic, geometric, and harmonic means of $(m,n)_b$. Additionally, we also
deduce asymptotic formulas with error terms for the means of $(n_1,n_2,\cdots,
n_k)_b$, and $[n_1,n_2,\cdots, n_k]_b$ over a set of lattice points, thereby
generalizing some of the previous work on gcd and lcm-sum estimates.
