Squaring both sides gives $a^{2}+b+2 a \sqrt{b}=15+\sqrt{216}$; separating rational from irrational parts, we get $a^{2}+b=15,4 a^{2} b=216$, so $a^{2}$ and $b$ equal 6 and $9 . a$ is an integer, so $a^{2}=9, b=6 \Rightarrow a / b=3 / 6=1 / 2$. (We cannot have $a=-3$, since $a+\sqrt{b}$ is positive.)