1. **State the problem:** We need to order the given expressions from least to greatest. 2. **List the expressions:** - $\frac{3\pi}{5}$ - $\infty$ - $\frac{6}{7}$ - $e^5$ - $\sum_{i=4}^7 i$ - $\sqrt{7}$ - $2!$ - $\int_3^8 x \, dx$ - $\log_4(11)$ 3. **Evaluate each expression:** - $\frac{3\pi}{5} = \frac{3 \times 3.1416}{5} = \frac{9.4248}{5} = 1.885$ (approx) - $\infty$ is infinitely large, so it is the greatest. - $\frac{6}{7} = 0.857$ (approx) - $e^5 = 148.413$ (approx) - $\sum_{i=4}^7 i = 4 + 5 + 6 + 7 = 22$ - $\sqrt{7} = 2.6458$ (approx) - $2! = 2 \times 1 = 2$ - $\int_3^8 x \, dx = \left[ \frac{x^2}{2} \right]_3^8 = \frac{8^2}{2} - \frac{3^2}{2} = \frac{64}{2} - \frac{9}{2} = 32 - 4.5 = 27.5$ - $\log_4(11) = \frac{\ln 11}{\ln 4} = \frac{2.3979}{1.3863} = 1.73$ (approx) 4. **Order from least to greatest:** - $\frac{6}{7} = 0.857$ - $2! = 2$ - $\sqrt{7} = 2.6458$ - $\log_4(11) = 1.73$ (Note: This is less than $\sqrt{7}$, so reorder) Reordering step 4 with correct order: - $\frac{6}{7} = 0.857$ - $2! = 2$ - $\log_4(11) = 1.73$ - $\sqrt{7} = 2.6458$ This shows $\log_4(11)$ is actually less than $2!$, so correct order is: - $\frac{6}{7} = 0.857$ - $\log_4(11) = 1.73$ - $2! = 2$ - $\sqrt{7} = 2.6458$ Continue ordering: - $\frac{3\pi}{5} = 1.885$ (which is between $\log_4(11)$ and $2!$, so reorder again) Final correct order: - $\frac{6}{7} = 0.857$ - $\log_4(11) = 1.73$ - $\frac{3\pi}{5} = 1.885$ - $2! = 2$ - $\sqrt{7} = 2.6458$ - $\sum_{i=4}^7 i = 22$ - $\int_3^8 x \, dx = 27.5$ - $e^5 = 148.413$ - $\infty$ 5. **Summary:** The squares in order from least to greatest are: $$\frac{6}{7} < \log_4(11) < \frac{3\pi}{5} < 2! < \sqrt{7} < \sum_{i=4}^7 i < \int_3^8 x \, dx < e^5 < \infty$$