Example 1.4.6
Use the order of operations to simplify the following expressions.

\(10+2\cdot 3\text{.}\) With this expression, we have the operations of addition and multiplication. The order of operations says the multiplication has higher priority, so execute that first:
\begin{align*} 10+2\cdot 3\amp =10+\nextoperation{2\cdot 3}\\ \amp=10+\highlight{6}\\ \amp=\highlight{16} \end{align*} 
\(4+10\div 2  1\text{.}\) With this expression, we have addition, division, and subtraction. According to the order of operations, the first thing we need to do is divide. After that, we'll apply the addition and subtraction, working left to right:
\begin{align*} 4+10\div21\amp=4+\nextoperation{10\div2}1\\ \amp=\nextoperation{4+\highlight{5}}1\\ \amp=\highlight{9}1\\ \amp=\highlight{8} \end{align*} 
\(710+4\text{.}\) This example only has subtraction and addition. While the acronym PEMDAS may mislead you to do addition before subtraction, remember that these operations have the same priority, and so we work left to right when executing them:
\begin{align*} 710+4\amp=\nextoperation{710}+4\\ \amp=\highlight{3}+4\\ \amp=1 \end{align*} 
\(20\div 4\cdot 7\text{.}\) This expression has only division and multiplication. Again, remember that although PEMDAS shows “MD,” the operations of multiplication and division have the same priority, so we'll apply them left to right:
\begin{align*} 20\div 4\cdot 5\amp=\nextoperation{20\div 4} \cdot 5\\ \amp=\highlight{5}\cdot5\\ \amp=\highlight{25} \end{align*} 
\((6+7)^2\text{.}\) With this expression, we have addition inside a set of parentheses, and an exponent of \(2\) outside of that. We must compute the operation inside the parentheses first, and after that we'll apply the exponent:
\begin{align*} (6+7)^2\amp= (\nextoperation{6+7})^2\\ \amp= \highlight{13}^2 \\ \amp= \highlight{169} \end{align*} 
\(4(2)^3\text{.}\) This expression has multiplication and an exponent. There are parentheses too, but no operation inside them. Parentheses used in this manner make it clear that the \(4\) and \(2\) are separate numbers, not to be confused with \(42\text{.}\) In other words, \(4(2)^3\) and \(42^3\) mean very different things. Exponentiation has the higher priority, so we'll apply the exponent first, and then we'll multiply:
\begin{align*} 4(2)^3 \amp= 4\nextoperation{(2)^3}\\ \amp= 4(\highlight{8})\\ \amp= \highlight{32} \end{align*}