1. **State the problem:** Approximate the value of $12^{3.14}$ using a calculator and round to three decimal places. 2. **Formula used:** To calculate $a^b$, where $a$ is the base and $b$ is the exponent, use the formula $$a^b = e^{b \ln(a)}$$ where $e$ is Euler's number and $\ln$ is the natural logarithm. 3. **Calculate intermediate values:** - Calculate $\ln(12)$. - Multiply $3.14 \times \ln(12)$. - Calculate $e^{3.14 \times \ln(12)}$. 4. **Perform the calculation:** - $\ln(12) \approx 2.4849$ - $3.14 \times 2.4849 = 7.7995$ - $e^{7.7995} \approx 2441.141$ 5. **Final answer:** $$12^{3.14} \approx 2441.141$$ This is the approximate value rounded to three decimal places.