Operations are things that you do with numbers. When you have two numbers, and you combine them to get a single number, you are performing an addition operation. If you take away a number from another, you are performing a subtraction operation. If you are doubling a number, you are performing a multiplication operation. If you are cutting a whole cake into smaller slices, you are performing a division operation.
When we are presented with an expression with several operations, the order of operations helps us to prioritize which one should be done first. If there are no rules to follow, it is possible to have different answers to a single problem. That is something we don't want to happen.
There is a mnemonic that will help you remember the order of operations. It's PEMDAS which stands for "Please Excuse My Dear Aunt Sally".
P - parenthesis
E - exponents
M - multiplication
D - division
A - addition
S - subtraction
The order of operations instructs us to do everything inside the parenthesis first. Next, simplify things as much as possible with exponents. Now, we divide or multiply from left to right, depending on which comes first. Finally, add or subtract from left to right, depending on which comes first.
Example 1: Simplify using the order of operations.
$$9 - 3 \times 2$$
Based on the order of operations, multiplication takes precedence before subtraction. That means we are going to multiply first then subtract.
$$\eqalign{
& 9 - 3 \times 2 \cr
& 9 - 6 \cr
& 3 \cr} $$
Example 2: Simplify using the order of operations.
$$\left( {4 + 10} \right) \div 2$$
Scanning from left to right, addition comes first before division. Although division takes priority over addition, the addition operation is inside the parenthesis therefore we should add the stuff inside the parenthesis first. Then we divide.
$$\eqalign{
& \left( {4 + 10} \right) \div 2 \cr
& 14 \div 2 \cr
& 7 \cr} $$
Example 3: Simplify using the order of operations.
$$9 - {2^3} \div 4$$
First we must deal with the exponent, then we divide, and finally we subtract.
$$\eqalign{
& 9 - {2^3} \div 4 \cr
& 9 - 8 \div 4 \cr
& 8 - 2 \cr
& 6 \cr} $$