Proving an absence of rational and integer solutions. Prove $$(x+y\sqrt2)^2+(z+t\sqrt2)^2=5+4\sqrt2$$
 has no solution in rational $(x,y,z,t)$
Prove $$(5+3\sqrt2)^m=(3+5\sqrt2)^n$$ 
has no solution for positive integers $(m,n)$
How do I approach these kinds of problems? I'm not sure where to start. Also, what are some more problems in this category to practice?

Hint for the second part: Take norms: $N(a+b\sqrt2)=a^2-2b^2$. The key property is $N(\alpha\beta)=N(\alpha)N(\beta)$.
