The general formula for arc length is:
Rewrite the original equation in $y =$ form.
$y=(1-x^{2/3})^{3/2}$
$y^\prime=\frac{3}{2}(1-x^{2/3})^{1/2}\big(-\frac{2}{3}x^{-1/3}\big)$
Before computing the arc length, graph this curve to determine the bounds of integration and use the symmetry of the graph.