Answer
The center of mass of the Earth-Moon system is a distance of $~~4.64\times 10^6~m~~$ from the center of the Earth.
Work Step by Step
The mass of the Earth is $5.98\times 10^{24}~kg$
The mass of the Moon is $7.36\times 10^{22}~kg$
Let the position of the center of the Earth be $x = 0$
Let the position of the center of the Moon be $x = 3.82\times 10^8~m$
We can find the center of mass of the Earth-Moon system:
$x_{com} = \frac{(5.98\times 10^{24}~kg)(0)+(7.36\times 10^{22}~kg)(3.82\times 10^8~m)}{5.98\times 10^{24}~kg+7.36\times 10^{22}~kg}$
$x_{com} = 4.64\times 10^6~m$
The center of mass of the Earth-Moon system is a distance of $~~4.64\times 10^6~m~~$ from the center of the Earth.
