Question: Work out the value of $10\ 000^{3/4}$
1. **State the problem:** We need to find the value of $$10\ 000^{3/4}$$.
2. **Recall the exponent rule:** For any positive number $a$ and rational exponent $m/n$, $$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$.
3. **Apply the rule:** Here, $a = 10\ 000$, $m = 3$, and $n = 4$. So,
$$10\ 000^{3/4} = \left(\sqrt[4]{10\ 000}\right)^3$$.
4. **Calculate the fourth root:** Since $10\ 000 = 10^4$,
$$\sqrt[4]{10\ 000} = \sqrt[4]{10^4} = 10$$.
5. **Raise to the power 3:**
$$10^3 = 10 \times 10 \times 10 = 1000$$.
6. **Final answer:**
$$10\ 000^{3/4} = 1000$$.