1. **State the problem:** Solve the equation $$10^{8t} \times 10^{2 - t} = 11$$ for $t$. 2. **Use the property of exponents:** When multiplying powers with the same base, add the exponents: $$10^{8t} \times 10^{2 - t} = 10^{8t + 2 - t} = 10^{7t + 2}$$ 3. **Rewrite the equation:** $$10^{7t + 2} = 11$$ 4. **Take the logarithm base 10 of both sides:** $$\log_{10}(10^{7t + 2}) = \log_{10}(11)$$ 5. **Use the logarithm power rule:** $$7t + 2 = \log_{10}(11)$$ 6. **Isolate $t$:** $$7t = \log_{10}(11) - 2$$ 7. **Divide both sides by 7:** $$t = \frac{\log_{10}(11) - 2}{7}$$ 8. **Calculate the numerical value:** $$\log_{10}(11) \approx 1.0414$$ $$t = \frac{1.0414 - 2}{7} = \frac{-0.9586}{7} \approx -0.1369$$ **Final answer:** $$t \approx -0.1369$$