Answer
The equation is symmetric about the $x$-axis, the $y$-axis and the origin.
Work Step by Step
$x^{2/3}+y^{2/3}=1$
Check for symmetry about the $y$-axis by substituting $x$ by $-x$ in the given expression and simplifying:
$(-x)^{2/3}+y^{2/3}=1$
$\sqrt[3]{(-x)^{2}}+y^{2/3}=1$
$\sqrt[3]{x^{2}}+y^{2/3}=1$
$x^{2/3}+y^{2/3}=1$
Since substituting $x$ by $-x$ yielded an equivalent expression, the equation is symmetric about the $y$-axis
Check for symmetry about the $x$-axis by substituting $y$ by $-y$ in the given expression and simplifying:
$x^{2/3}+(-y)^{2/3}=1$
$x^{2/3}+\sqrt[3]{(-y)^{2}}=1$
$x^{2/3}+\sqrt[3]{y^{2}}=1$
$x^{2/3}+y^{2/3}=1$
Since substituting $y$ by $-y$ yielded an equivalent expression, the equation is symmetric about the $x$-axis
Since the equation is symmetric to both the $x$ and $y$-axis, it is also symmetric about the origin.
The graph is shown in the answer section.
