# How do you determine if f(x,y)=(x^2y^2)/sqrt(x^2+y^2) is homogeneous and what would it's degree be?

$f \left(x , y\right)$ is homogeneous of degree 3
$f \left(\alpha x , \alpha y\right) = \frac{{\left(\alpha x\right)}^{2} {\left(\alpha y\right)}^{2}}{\sqrt{{\left(\alpha x\right)}^{2} + {\left(\alpha y\right)}^{2}}} = \frac{{\alpha}^{4} {x}^{2} {y}^{2}}{\sqrt{{\alpha}^{2} \left({x}^{2} + {y}^{2}\right)}} = \frac{{\alpha}^{4} {x}^{2} {y}^{2}}{\alpha \sqrt{{x}^{2} + {y}^{2}}} = {\alpha}^{3} f \left(x , y\right)$
so we can see that $f \left(x , y\right)$ is homogeneous of degree 3