To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
order of operations
The order of operations is a set of rules used in mathematics to determine the sequence in which operations should be performed to correctly solve an expression.
The standard order in which these operations must be carried out is often remembered by the acronym PEMDAS:
P(arentheses): First, perform all operations inside parentheses ( ) or other grouping symbols like brackets [ ] and braces { }. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols.
E(xponents): Next, solve any exponents ([latex]a^n = a \cdot a \cdot ... \cdot a[/latex]) and roots.
M(ultiplication) and D(ivision): Then, perform all multiplication and division from left to right, as they appear in the expression. These operations are of equal precedence and are carried out in the order they occur from left to right.
A(ddition) and S(ubtraction): Lastly, perform all addition and subtraction from left to right, as they appear. Like multiplication and division, these operations are of equal precedence and should be carried out from left to right.
This order of operations is true for all real numbers.
How to: Given a mathematical expression, simplify it using the order of operations.
Simplify any expressions within grouping symbols.
Simplify any expressions containing exponents or radicals.
Perform any multiplication and division in order, from left to right.
Perform any addition and subtraction in order, from left to right.
Simplify the expression [latex]8+(3\cdot2^2)-3[/latex].
Parentheses: First, solve the expression inside the parentheses.
Inside the parentheses: [latex]3\cdot 2^2[/latex]
Exponents: Solve any exponents.
[latex]2^2 = 2 \cdot 2 = 4[/latex], so the expression inside the parentheses becomes [latex]3\cdot 4[/latex].
Multiplication: Then, perform the multiplication.
[latex]3\cdot4 =12[/latex], so now the expression is [latex]8+12-3[/latex].
Addition and Subtraction: Finally, perform the addition and subtraction from left to right.
[latex]8+12-3[/latex]
[latex]20-3[/latex]
[latex]17[/latex]
Therefore, [latex]8+(3\cdot2^2)-3 = 17[/latex].
For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.
Use the order of operations to evaluate each of the following expressions.
Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
[latex]\begin{align}\frac{14-3\cdot2}{2\cdot5-3^{2}} & =\frac{14-3\cdot2}{2\cdot5-9} && \text{Simplify exponent} \\ & =\frac{14-6}{10-9} && \text{Simplify products} \\ & =\frac{8}{1} && \text{Simplify quotient} \\ & =8 && \text{Simplify quotient}\end{align}[/latex]
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.