Explain why x2 y2 30 not having any ration solutions implies

Explain why x² +y² -3=0 not having any ration solutions implies x²+y²-3^k=0 has no rational solutions for k an odd, positive integer.

Let, x^2+y^2=3 have no rational solutions

We prove by contradiction that

x^2+y^2=3^k has no rational solutions

Assume:

x^2+y^2=3^k has rational solutions,u,v

k is odd ie k=2m+1 for some integer m

u^2+v^2=3^{2m+1}

u^2+v^2=3^{2m}*3

u^2/3^{2m}+v^2/3^{2m}=3

(u/3^m)^2+(v/3^m)^2=3

u,v are rational. Hence, u/3^m and v/3^m are rational

Hence, u/3^m and v/3^m are rational solutions to

x^2+y^2=3

which is a contradiction

HEnce there are no rational solutoins to

x^2+y^2=3^k, whre k is odd positive intger