Planet a and planet b are in circular orbits around a distant star. planet a is 7.8 times farther from the star than is planet
b. what is the ratio of their speeds va/vb?

To find the ratio of planetary speeds Va/Vb we need the orbital velocity formula: 

V=√({G*M}/R), where G is the gravitational constant, M is the mass of the distant star and R is the distance of the planet from the star it is orbiting. 

So Va/Vb=[√( {G*M}/Ra) ] / [√( {G*M}/Rb) ], in our case Ra = 7.8*Rb 

Va/Vb=[ √( {G*M}/{7.8*Rb} ) ]  / [√( {G*M}/Rb )], we put everything under one square root by the rule: (√a) / (√b) = √(a/b) 

Va/Vb=√ [ { (G*M)/(7.8*Rb) } / { (G*M)/(Rb) } ], when we cancel out G, M and Rb we get:

Va/Vb=√(1/7.8)/(1/1)=√(1/7.8)=0.358 so the ratio of Va/Vb = 0.358.