1. **State the problem:** We need to order the given expressions from greatest to least, including approximate values. 2. **List the expressions:** - $\sqrt{7}$ - $6!$ - $e^2$ - $\sum_{i=2}^6 i$ - $\frac{12}{13}$ - $\frac{3\pi}{4}$ - $\log_3(10)$ - $\infty$ - $\int_3^6 x \, dx$ 3. **Calculate or approximate each value:** - $\sqrt{7} \approx 2.64575$ - $6! = 720$ - $e^2 \approx 7.38906$ - $\sum_{i=2}^6 i = 2+3+4+5+6 = 20$ - $\frac{12}{13} \approx 0.92308$ - $\frac{3\pi}{4} = 3 \times \frac{3.14159}{4} \approx 2.35619$ - $\log_3(10) = \frac{\ln 10}{\ln 3} \approx \frac{2.30259}{1.09861} \approx 2.09590$ - $\infty$ is infinite, the greatest value - $\int_3^6 x \, dx = \left[ \frac{x^2}{2} \right]_3^6 = \frac{6^2}{2} - \frac{3^2}{2} = \frac{36}{2} - \frac{9}{2} = 18 - 4.5 = 13.5$ 4. **Order from greatest to least:** - $\infty$ - $6! = 720$ - $e^2 \approx 7.38906$ - $\sum_{i=2}^6 i = 20$ - $\int_3^6 x \, dx = 13.5$ - $\sqrt{7} \approx 2.64575$ - $\frac{3\pi}{4} \approx 2.35619$ - $\log_3(10) \approx 2.09590$ - $\frac{12}{13} \approx 0.92308$ 5. **Final answer:** $$\infty > 720 > 20 > 13.5 > 7.38906 > 2.64575 > 2.35619 > 2.09590 > 0.92308$$ Note: The order of $20$ and $13.5$ was corrected to $20 > 13.5$ since $20$ is greater than $13.5$.