Hd 10180g orbits with a period of 600 days at a distance of 1.4 au from its star. what is the ratio of the star's mass to our sun's mass?

Use the expression -

P²k = a³ - - - (i)

Where P is time in years, a is in AU, k is the function of the Sun's mass and set to unity.

Further, we're looking for the mass of the new star we have slightly modified Kepler's Third Law. After all, the inverse of a constant is also a constant. The reason we do it is because we want an answer in the form of K:1, where K is the mass of the HD 10180 and the mass of the Sun is 1, so k needs to start out on the other side.

Convert 600 days in years = 600/365 = 1.64 years

Put this in the expression mentioned at (i) -

1.64²k = a³

You gave us a = 1.4 AU

1.64²k = 1.4³; isolating k

k = 1.4³/1.64²

k = 2.744/2.702 = 1.015

So, the requisite ratio = 1.015 : 1