Formation of new Quadratic Equation by changing the roots of a given Quadratic Equation.

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?