Answer
The ratio of the moon's orbital speed at perigee to that at apogee is $1.14$
Work Step by Step
We can use conservation of angular momentum to find the ratio of the orbital speeds:
$L_p = L_a$
$m~v_p~r_p = m~v_a~r_a$
$\frac{v_p}{v_a} = \frac{r_a}{r_p}$
$\frac{v_p}{v_a} = \frac{4.07\times 10^5~km}{3.56\times 10^5~km}$
$\frac{v_p}{v_a} = 1.14$
The ratio of the moon's orbital speed at perigee to that at apogee is $1.14$.
