Answer
See answer.
Work Step by Step
We are asked to use Kepler’s third law to find the orbital radius of each moon, using Io’s data.
$$(\frac{r_1}{r_2})^3=(\frac{T_1}{T_2})^2$$
$$r_1=r_2 (\frac{T_1}{T_2})^{2/3}$$
Now apply this to the 3 moons other than Io.
$$r_{Europa}=r_{Io} (\frac{T_{Europa}}{T_{Io}})^{2/3}$$
$$r_{Europa}=(422\times10^3 km) (\frac{3.55d}{1.77d})^{2/3}=671\times10^3 km $$
$$r_{Ganymede}=r_{Io} (\frac{T_{ Ganymede }}{T_{Io}})^{2/3}$$
$$r_{ Ganymede }=(422\times10^3 km) (\frac{7.16d}{1.77d})^{2/3}=1070\times10^3 km $$
$$r_{Callisto}=r_{Io} (\frac{T_{ Callisto }}{T_{Io}})^{2/3}$$
$$r_{ Callisto }=(422\times10^3 km) (\frac{16.7d}{1.77d})^{2/3}=1880\times10^3 km $$
These numbers agree very well with the tabulated values.
