If $\alpha$ and $\beta$ are the root of equation $ax^2+bx+c=0$.
Prove that the equation whose root is $\alpha^n$ and $\beta^n$ is $$a(x^\frac 1n)^2+b(x^\frac 1n)+c=0$$
I had already found the equation whose root is whose root is $\alpha^n$ and $\beta^n$ by using $$x^2-(\text{sum of roots})x+\text{product of roots}\implies x^2-(\alpha^n+\beta^n)x+\alpha^n\beta^n=0$$
Is this equation is same as what is given to prove?
You are trying to find the new quadratic equation whose roots are $\alpha^n$ and $\beta^n$ your second equation is OK and it does the required. But the method of transformation of equation will not work, because your first equation when rationalized for an equation with rational powers either does not get rationalized or it becomes an equation of much higher degree which is definitely not a quadratic. So your first method would work only if new roots are $\alpha^2$ and $\beta^2$. You may check that you cannot construct a new quadratic equation whose roots are $\alpha^3$ and $\beta^3$.