1. **State the problem:** Determine whether each expression is equivalent to $7^4$, $7^{-4}$, or Neither. 2. **Recall the rules:** - $a^m \cdot a^n = a^{m+n}$ - $(a^m)^n = a^{m \cdot n}$ - $\frac{1}{a^m} = a^{-m}$ - $\left(\frac{1}{a^m}\right)^{-1} = a^m$ 3. **Evaluate each expression:** **a.** $\frac{1}{7^{-4}} = 7^{4}$ because $\frac{1}{7^{-4}} = 7^{4}$ by the negative exponent rule. **b.** $7^{2} \cdot 7^{-2} = 7^{2 + (-2)} = 7^{0} = 1$ which is neither $7^{4}$ nor $7^{-4}$. **c.** $(7^{2})^{-2} = 7^{2 \cdot (-2)} = 7^{-4}$. **d.** $\left(\frac{1}{7^{4}}\right)^{-1} = (7^{-4})^{-1} = 7^{4}$. 4. **Summary:** - a is equivalent to $7^{4}$ - b is Neither - c is equivalent to $7^{-4}$ - d is equivalent to $7^{4}$ **Final answer:** - a: $7^{4}$ - b: Neither - c: $7^{-4}$ - d: $7^{4}$