1. The problem is to arrange the given mathematical expressions from smallest to biggest. 2. Let's evaluate or understand each expression: - $\int_2^7 x \, dx$: This is the definite integral of $x$ from 2 to 7. - $\log_2(7)$: Logarithm base 2 of 7. - $e^6$: Exponential function with base $e$ raised to 6. - $\sqrt{10}$: Square root of 10. - $3!$: Factorial of 3. - $\infty$: Infinity, which is larger than any finite number. - $\frac{5}{9}$: Fraction 5 divided by 9. - $\sum_{i=4}^8 i$: Sum of integers from 4 to 8. - $\frac{5\pi}{3}$: Five times pi divided by 3. 3. Calculate each value: - $\int_2^7 x \, dx = \left[ \frac{x^2}{2} \right]_2^7 = \frac{7^2}{2} - \frac{2^2}{2} = \frac{49}{2} - \frac{4}{2} = \frac{45}{2} = 22.5$ - $\log_2(7) = \frac{\ln 7}{\ln 2} \approx \frac{1.9459}{0.6931} \approx 2.807$ - $e^6 \approx 403.429$ - $\sqrt{10} \approx 3.162$ - $3! = 3 \times 2 \times 1 = 6$ - $\infty$ is infinite, so it is the largest. - $\frac{5}{9} \approx 0.555$ - $\sum_{i=4}^8 i = 4 + 5 + 6 + 7 + 8 = 30$ - $\frac{5\pi}{3} \approx \frac{5 \times 3.1416}{3} = 5.236$ 4. Now, arrange from smallest to biggest: $\frac{5}{9} \approx 0.555 < \log_2(7) \approx 2.807 < \sqrt{10} \approx 3.162 < 3! = 6 < \frac{5\pi}{3} \approx 5.236$ (note $5.236 < 6$ so correct order is $5.236 < 6$ so swap) Correct order: $\frac{5}{9} < \log_2(7) < \sqrt{10} < \frac{5\pi}{3} < 3! < \int_2^7 x \, dx < \sum_{i=4}^8 i < e^6 < \infty$ Numerically: $0.555 < 2.807 < 3.162 < 5.236 < 6 < 22.5 < 30 < 403.429 < \infty$ 5. Explanation: We evaluated each expression to a decimal or exact value to compare their sizes. Infinity is always the largest. Final answer: $\frac{5}{9} < \log_2(7) < \sqrt{10} < \frac{5\pi}{3} < 3! < \int_2^7 x \, dx < \sum_{i=4}^8 i < e^6 < \infty$