If the coefficient of $x^{15}$ in the expansion of $\left(a x^{3}+\dfrac{1}{b x^{1/3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1/3}-\dfrac{1}{b x^{3}}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$: