Order Expressions 4B7E22

1. **State the problem:** We need to order the following expressions from least to greatest: $$\frac{8}{9}, \infty, \int_1^4 x\,dx, \frac{6\pi}{6}, e^4, \log_2(20), \sum_{i=1}^5 i, 2!, \sqrt{21}$$ 2. **Calculate each expression:** - $\frac{8}{9} \approx 0.8889$ - $\infty$ is infinitely large, so it will be the greatest. - $\int_1^4 x\,dx = \left[ \frac{x^2}{2} \right]_1^4 = \frac{4^2}{2} - \frac{1^2}{2} = \frac{16}{2} - \frac{1}{2} = \frac{15}{2} = 7.5$ - $\frac{6\pi}{6} = \pi \approx 3.1416$ - $e^4 \approx 54.5982$ - $\log_2(20)$ means the logarithm base 2 of 20. Since $2^4=16$ and $2^5=32$, $\log_2(20)$ is between 4 and 5. More precisely, $\log_2(20) \approx 4.3219$ - $\sum_{i=1}^5 i = 1+2+3+4+5 = 15$ - $2! = 2 \times 1 = 2$ - $\sqrt{21} \approx 4.5826$ 3. **Order the values:** $0.8889 < 2 < 3.1416 < 4.3219 < 4.5826 < 7.5 < 15 < 54.5982 < \infty$ 4. **Match to original expressions:** - $\frac{8}{9} \approx 0.8889$ - $2! = 2$ - $\frac{6\pi}{6} = \pi \approx 3.1416$ - $\log_2(20) \approx 4.3219$ - $\sqrt{21} \approx 4.5826$ - $\int_1^4 x\,dx = 7.5$ - $\sum_{i=1}^5 i = 15$ - $e^4 \approx 54.5982$ - $\infty$ is the greatest **Final order from least to greatest:** $$\frac{8}{9} < 2! < \frac{6\pi}{6} < \log_2(20) < \sqrt{21} < \int_1^4 x\,dx < \sum_{i=1}^5 i < e^4 < \infty$$