$\text{First, check if it is even:}$

$f(a)=a^{\frac{1}{3}}$

$f(-a)=(-a)^{\frac{1}{3}}=-a^{\frac{1}{3}}$

$\text{since }f(a) \neq f(-a),$

$f(x)\text{ IS NOT EVEN!}$

$\text{Next, check if it is odd:}$

$f(-a)=(-a)^{\frac{1}{3}}=-a^{\frac{1}{3}}$

$-f(a)=-a^{\frac{1}{3}}$

$\text{since }f(-a)= -f(a),{\ \ \ \ \ }f(x)\text{ IS ODD!}$