Question: Write $\frac{1}{\sqrt[3]{13}}$ in the form $13^t$. What is the value of $t$? Give your answer in its simplest form.
1. **State the problem:** We want to express $\frac{1}{\sqrt[3]{13}}$ in the form $13^t$ and find the value of $t$.
2. **Recall the rules:** The cube root of 13 can be written as an exponent: $\sqrt[3]{13} = 13^{\frac{1}{3}}$.
3. **Rewrite the expression:**
$$\frac{1}{\sqrt[3]{13}} = \frac{1}{13^{\frac{1}{3}}}$$
4. **Use the rule for negative exponents:**
$$\frac{1}{a^b} = a^{-b}$$
5. **Apply this rule:**
$$\frac{1}{13^{\frac{1}{3}}} = 13^{-\frac{1}{3}}$$
6. **Conclusion:** The expression $\frac{1}{\sqrt[3]{13}}$ can be written as $13^t$ where
$$t = -\frac{1}{3}$$