System of Exponential Equations

Divide the two equations: $\frac{x^{2018} y^{2017}}{x^{2016} y^{2019}} = \frac{1/2}{8} \Rightarrow \frac{x^2}{y^2} = \frac{1}{16} \Rightarrow \frac{x}{y} = \pm \frac{1}{4}$. $x^2 = \frac{y^2}{16}$. Multiply equations: $(x^{2018} y^{2017})(x^{2016} y^{2019}) = 4 \Rightarrow x^{4034} y^{4036} = 4 \Rightarrow (x y)^{4034} y^2 = 4$. Or rewrite: $(x^2)^{1009} (y^3)^{672} \dots$ Better approach: From $x^2 = y^2 / 16 \Rightarrow x = y/4$ or $x = -y/4$. Substitute $x = y/4$ into $x^{2016} y^{2019} = 8$: $(y/4)^{2016} y^{2019} = 8 \Rightarrow y^{4035} = 8 \cdot 4^{2016} = 2^3 \cdot 2^{4032} = 2^{4035} \Rightarrow y = 2$. Then $y^3 = 8$. Since $y = 2$, $x^2 = 2^2 / 16 = 4/16 = 1/4$. $x^2 + y^3 = 1/4 + 8 = 33/4$.