1. **State the problem:** We need to find values of $n$ and $d$ such that $$18^{\frac{9}{8}} = (\sqrt[n]{18})^d.$$\n\n2. **Recall the formula:** The $n$th root of a number $a$ can be written as a fractional exponent: $$\sqrt[n]{a} = a^{\frac{1}{n}}.$$\n\n3. **Rewrite the right side:** Using the exponent rule, $$(\sqrt[n]{18})^d = \left(18^{\frac{1}{n}}\right)^d = 18^{\frac{d}{n}}.$$\n\n4. **Set the exponents equal:** Since the bases are the same (18), the exponents must be equal: $$\frac{9}{8} = \frac{d}{n}.$$\n\n5. **Solve for $d$ in terms of $n$:** $$d = \frac{9}{8} n.$$\n\n6. **Choose integer values:** To keep $d$ and $n$ as integers, pick $n=8$, then $$d = \frac{9}{8} \times 8 = 9.$$\n\n7. **Final answer:** $$n=8, \quad d=9.$$