Show that following subset of $\mathbb R^2$ is compact

Show that following subset of $\mathbb R^2$ is compact

Show that following subset of $\mathbb R^2$ is compact: $$\{(x, y) :
x^{2/3}+y^{2/3} = 1\}$$
My attempt:
$A=\{(x, y) : x^{2}+y^{2} = 1\}$ is compact,
$f:(x,y)\mapsto(x^{1/3},y^{1/3})$ is continuous and $f(A)=\{(x, y) :
x^{2/3}+y^{2/3} = 1\}.$ So $f(A)$ is compact.
Am I correct?