1. **State the problem:** Solve the equation $$(2^4)^{t-2} \times 2^{2t} = (2^2)^{t+2}$$ for $t$. 2. **Recall exponent rules:** - Power of a power: $$(a^m)^n = a^{mn}$$ - Product of powers with the same base: $$a^m \times a^n = a^{m+n}$$ 3. **Apply the power of a power rule:** $$(2^4)^{t-2} = 2^{4(t-2)} = 2^{4t - 8}$$ $$(2^2)^{t+2} = 2^{2(t+2)} = 2^{2t + 4}$$ 4. **Rewrite the equation using these simplifications:** $$2^{4t - 8} \times 2^{2t} = 2^{2t + 4}$$ 5. **Use the product of powers rule on the left side:** $$2^{(4t - 8) + 2t} = 2^{2t + 4}$$ $$2^{6t - 8} = 2^{2t + 4}$$ 6. **Since the bases are equal, set the exponents equal:** $$6t - 8 = 2t + 4$$ 7. **Solve for $t$:** $$6t - 8 = 2t + 4$$ $$6t - \cancel{8} + \cancel{8} = 2t + 4 + 8$$ $$6t = 2t + 12$$ $$6t - 2t = 12$$ $$4t = 12$$ $$t = \frac{12}{4}$$ $$t = 3$$ **Final answer:** $$t = 3$$