Question: Work out the value of $81^{3/2}$
1. **State the problem:** We need to find the value of $81^{3/2}$.
2. **Recall the exponent rule:** For any positive number $a$ and rational exponent $m/n$, we have $$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}.$$ Here, $m=3$ and $n=2$.
3. **Apply the rule:** $$81^{3/2} = \left(\sqrt{81}\right)^3 = (9)^3.$$
4. **Calculate the square root:** $\sqrt{81} = 9$ because $9 \times 9 = 81$.
5. **Calculate the cube:** $9^3 = 9 \times 9 \times 9 = 729$.
6. **Final answer:** $$\boxed{729}.$$