The order of operations is a set of rules that have been standardized and agreed upon regarding how to carry out mathematical calculations.

## Rules of Order of Operations

1. Perform all operations that are within the grouping symbols starting from the innermost. Examples of grouping symbols are parentheses ( ) and brackets [ ].

2. Evaluate or simplify the expressions that contain exponents.

3. Multiply or divide depending on which operation comes first from left to right.

4. Add or subtract depending on which operation comes first from left to right.

## Example of Order of Operations

Example 1: \(11 - 2 \times 3\)

There are no grouping symbols and exponents. From left to right, we have subtraction and multiplication. Since multiplication has a higher priority than subtraction, we will multiply first and then subtract.

\(\eqalign{

& 11 - 2 \times 3 \cr

& 11 - 6 \cr

& 5 \cr} \)

Example 2: \(5 \times 2 + 12 \div 4\)

If we scan from left to right, we see multiplication, addition, and division. Multiplication and division are more important than addition. We will multiply first before we divide because multiplication comes first when viewed from left to right.

\(\eqalign{

& 5 \times 2 - 12 \div 4 \cr

& 10 - 12 \div 4 \cr} \)

We are left with subtraction and division. We should divide first because it has a higher priority than subtraction. Finally, finish it off with subtraction.

\(\eqalign{

& 10 - 12 \div 4 \cr

& 10 - 3 \cr

& 7 \cr} \)

Example 3: \(\left( {52 - 7} \right) \div 3 - {2^2}\)

At first glance, we see that there are parenthesis and exponents. Let's simplify the stuff inside the parenthesis and the exponential expression.

Note: \(52-7=45\) and \({2^2} = 4\)

Thus, we have

\(45 \div 3 - 4\)

Finally, we divide before subtracting.

\(\eqalign{

& 45 \div 3 - 4 \cr

& 15 - 4 \cr

& 11 \cr} \)