The PEMDAS Rule: Understanding Order of Operations

We are aware of the four basic mathematical operators which are addition, subtraction, multiplication and division. However, it is equally important to know the order in which the mathematical operations need to be done in expressions. This is where PEMDAS rule comes into play. For example, an expression like 2+3*4 can have different answers if operations are changed while keeping everything else the same. So let us see what the right order of operations is.

In any expression, we can have addition, subtraction, multiplication, long division, and exponents like a square root. It is also possible to have brackets or parenthesis in the equation. These brackets or parenthesis have a special position as in the order of operations parenthesis comes first. So, whatever is inside the brackets or parentheses solve them first. For example,

= (9-9) * 1 (FIRST : solve the parenthesis) = 0*1 = 0

Next, solve any exponential values in the equation, for example.

22 +3 = 4 + 3 = 7 (SECOND : solve the exponent)

Next, if there are any multiplications or divisions in the expression, solve them depending upon which comes first when reading the expression left to right. For example,

= (9-5) / 2 * 1 + 22 +3 (FIRST : solve the parenthesis)

= 4 / 2 * 1 + 4 +3 (Second : solve the exponent)

= 2 * 1 + 4 + 3 ( THIRD : solve multiplication or division whicever comes first from left)

= 0 + 4 +3 (FOURTH : solve the multiplication) = 7

After we have solved the parenthesis and exponents and set of division and multiplication depending upon whichever comes first, next if there are any additions or subtractions in the expression, we solve them next again depending upon which comes first from the left. For example,

= (9-9) * 1 – 3 + 22 / 4 + 3 (FIRST : solving the parenthesis)

= 0 * 1 – 3 + 22 / 4 + 3 (Second : solve the exponent)

= 0 * 1 – 3 + 4 / 4 + 3 (THIRD : solve the multiplication)

= 0 – 3 + 1 + 3 (FOURTH : solve the division)

= 0 – 3 + 4 (FIFTH : solve the subtraction)

= 1 (SIXTH: solve the addition)

So to sum it up below is the summary, the order in which the operations are to be done are as below:

  • Parentheses: like () or braces {} or brackets like [] and fraction bars.
  • Exponent also includes fractional exponents like roots.
  • Multiplication
  • Division
  • Addition
  • Subtraction

If we catch the first letter of each of the above, we get P E M D A S.

So, PEMDAS is the name of the rule used to know the order of operations. It also is convenient to remember this short abbreviation to remember the order of operations.

Also, do remember that even though in PEMDAS, M comes before D, one needs to make sure to solve the operation that comes first from the left.

For example, in the below expression if we happen to multiply before division, the answer would be incorrect. Since division comes first, we need to divide first.

= 10 / 5 * 4 + 1 = 2 * 4 + 1 = 8

Let us solve some expressions using this PEMDAS rule.

Example 1: Solve the following expression: 4 + (3 – 2)2 x 4 Ă· 2 – 1

Solution: Begin by the parentheses; (3 – 2) = 1

  1. Proceed to the exponential operation – 12 = 1
  2. Now we are left with; 4 + 1 x 4 ÷ 2 – 1 = ?
  3. Perform the multiplication and division, starting from left to right 1 x 4 = 4, 4 + 4 ÷ 2 – 1

Starting from the right;

  • 4 Ă· 2 = 2
  • 4 + 2 – 1 = ?
  • 4 + 2 = 6
  • 6 – 1 = ?
  • 6 – 1 = 5

Example 2: Let us solve (4 – 2) 2 – 1 x 2

Solution:

Start by opening the parentheses = (2) 2  1 x 2, Calculate the exponent = 4 – 1 x 2, Now do the multiplication = 4 – 2

Finally, do the subtraction and get the answer as 2. 

These examples help us understand how to handle the order of operations using the elegant rule of PEMDAS. Visit Cuemath for more information and practice worksheets related to PEMDAS.