1. Problem statement. Select the nine squares and order them from least to greatest among the expressions $\frac{5\pi}{4},\ e^{6},\ \log_{3}(3),\ \sum_{i=5}^{7} i,\ \int_{3}^{8} x\,dx,\ \infty,\ \frac{11}{16},\ 4!,\ \sqrt{21}$. 2. Formulas and key rules to use. - For logarithms use $\log_{a}(a)=1$ for any base $a>0$ with $a\neq 1$. - For a finite sum use definition $\sum_{i=m}^{n} a_i= a_m+\cdots+a_n$. - For factorials use $n!=n\times(n-1)\times\cdots\times1$. - For definite integrals use the Fundamental Theorem of Calculus: if $F'(x)=f(x)$ then $\int_{a}^{b} f(x)\,dx=F(b)-F(a)$. - Infinity $\infty$ is larger than every real number. 3. Evaluate each expression step by step and show intermediate work. - Evaluate $\dfrac{11}{16}$. $\dfrac{11}{16}=0.6875$. - Evaluate $\log_{3}(3)$. $\log_{3}(3)=1$. - Evaluate $\dfrac{5\pi}{4}$. $\dfrac{5\pi}{4}=1.25\pi\approx 3.926990716\,$. - Evaluate $\sqrt{21}$. $\sqrt{21}\approx 4.582575695\,$. - Evaluate $\sum_{i=5}^{7} i$. $\sum_{i=5}^{7} i=5+6+7=18$. - Evaluate $4!$. $4!=4\times3\times2\times1=24$. - Evaluate $\int_{3}^{8} x\,dx$ using antiderivative $\frac{x^{2}}{2}$. $$\int_{3}^{8} x\,dx=\left. \frac{x^{2}}{2} \right|_{3}^{8}=\frac{8^{2}-3^{2}}{2}=\frac{64-9}{2}. $$ Show intermediate simplification for the term $\dfrac{64}{2}$ by factoring and canceling common factor 2. $$\frac{64}{2}=\frac{2\cdot 32}{2}=\frac{\cancel{2}\cdot 32}{\cancel{2}}=32.$$ Continue the evaluation. $$\frac{64-9}{2}=\frac{64}{2}-\frac{9}{2}=32-\frac{9}{2}=\frac{55}{2}=27.5.$$ - Evaluate $e^{6}$. $e^{6}\approx 403.428793492735$. - Interpret $\infty$. $\infty$ denotes an unbounded quantity larger than any real number. 4. Summary of numeric values (approximate where needed). - $\dfrac{11}{16}=0.6875$. - $\log_{3}(3)=1$. - $\dfrac{5\pi}{4}\approx 3.92699$. - $\sqrt{21}\approx 4.58258$. - $\sum_{i=5}^{7} i=18$. - $4!=24$. - $\int_{3}^{8} x\,dx=\dfrac{55}{2}=27.5$. - $e^{6}\approx 403.42879$. - $\infty$ is largest. 5. Order from least to greatest and final answer. Putting the evaluated values in order from least to greatest gives: $\dfrac{11}{16},\ \log_{3}(3),\ \dfrac{5\pi}{4},\ \sqrt{21},\ \sum_{i=5}^{7} i,\ 4!,\ \int_{3}^{8} x\,dx,\ e^{6},\ \infty$.