Nested everything

The first equation is equivalent to $x^3=3$ , or $x=\sqrt[3]{3}$ . The second is equivalent to $y=\sqrt{4y}\rightarrow y^2=4y\rightarrow y=4$ . The third is equivalent to $z=\sqrt{5+z}\rightarrow z^2=5+z \rightarrow z=\frac{1+\sqrt{21}}{2}$ . Thus, $x^3+y+z=3+4+\frac{1+\sqrt{21}}{2}=\frac{15+\sqrt{21}}{2}$ , so $\frac{a+b}{3c}=\frac{15+21}{3\times2}=\boxed{6}$ .