Section 1.3 Order of Operations
We combine real numbers using arithmetic operations such as addition, subtraction, multiplication, and division. If we want to carry out many such operations in a single expression, we must agree upon the rules on the correct order to carry this out to ensure we all obtain the same result. The conventional order of operations is referred to as PEMDAS.
Principle 1.9. Order of Operations.
Work from left to right evaluating the expression respecting the following order:
Parentheses: Perform all calculations inside parentheses or brackets (if parentheses are nested, work from inside out);
Exponents: Evaluate all exponential expressions \(a^b\text{;}\)
Multiplication and Division: Evaluate all products and quotients;
Addition and Subtraction: Evaluate all sums and differences.
Example 1.10.
Evaluate the following expressions. Show your work carefully as a sequence of equal expressions separated by the equal sign until complete. Then compare with the solution.
\(\displaystyle \displaystyle (-2+4)^2-3\cdot 4\)
\(\displaystyle \displaystyle -2^2-(-3)^2\div 4\)
\(\displaystyle \displaystyle -\frac{2^3+(-2)^3}{2^2+(-2)^2}\)
Solution.
\begin{align*}
(-2+4)^2-3\cdot 4 \amp= 2^2-3\cdot 4\\
\amp= 4-3\cdot 4\\
\amp= 4-12\\
\amp= \boxed{-8}
\end{align*}
\begin{align*}
-2^2-(-3)^2\div 4 \amp= 4 - 9 \div 4\\
\amp= 4 - \frac{9}{4}\\
\amp= \frac{16}{4} - \frac{9}{4}\\
\amp= \frac{16-9}{4}\\
\amp= \boxed{\frac{7}{4}}
\end{align*}
\begin{align*}
-\frac{2^3+(-2)^3}{2^2+(-2)^2}\amp= -\frac{8-8}{4+4}\\
\amp= -\frac{0}{8}\\
\amp= -0\\
\amp=\boxed{0}
\end{align*}
Checkpoint 1.11.
Evaluate without the use of a calculator
\begin{equation*}
\displaystyle 2 - \frac{2}{(\frac{1}{2}-1)^2}.
\end{equation*}
Show your work as a sequence of equal expressions separated by the equal sign “\(=\)”.
