Degree of the polynomial 

${\left(x^2-\sqrt{1-x^3}\right)}^4+{\left(x^2+\sqrt{1-x^3}\right)}^4$

We have 

$(x-y{)}^4+(x+y{)}^4=2\left[{ }^4{C}_0{x}^4{+}^4{C}_2{x}^2{y}^2{+}^4{C}_4{y}^4\right]$

Thus, $\begin{array}{l}{\left(x^2-\sqrt{1-x^3}\right)}^4+{\left(x^2+\sqrt{1-x^3}\right)}^4\\ =2\left[{ }^4{C}_0{\left(x^2\right)}^4{+}^4{C}_2{\left(x^2\right)}^2\left(1-x^3\right){+}^4{C}_4{\left(1-x^3\right)}^2\right]\end{array}$

which is a polynomial of degree$8.$