What is the PEMDAS Rule?
A math expression having multiple mathematical operators can be solved in different ways, and each method can give a different result. The PEMDAS rule defines a standard order of operations to be followed while evaluating an arithmetic expression. The word ‘PEMDAS’ is an acronym for ‘Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
Multiplication and division can be performed interchangeably, or they can be performed in a single step. The same is applicable for addition and subtraction. But addition and subtraction should not be performed before multiplication and division, and the latter should not be performed before simplifying the exponents. We can only break this order for simplifying expressions inside the parentheses. If there are parentheses in the expression, we need to simplify the terms inside them by applying the same rule.
Consider the expression (4 + 8) x 5 + 9 – 12\(\div \sqrt{4}\).
According to the PEMDAS rule, if an expression has multiple operators, we need to simplify the parentheses first, evaluate the part that has exponents, perform multiplication and division, and finally perform addition and subtraction. Let’s evaluate the expression using the rule.
(4 + 8) x 5 + 9 – 12\(\div \sqrt{4}\)
= 12 x 5 + 9 – 12 \(\div \sqrt{4}\) [Evaluate the expression inside parentheses]
Since \(\sqrt{4}\) is the same as \(4^{\frac{1}{2}}\), it is an exponent. So, it has to be simplified next.
= 12 x 5 + 9 – 12 \(\div\)2 [Evaluate the exponents and roots]
Next, we should perform multiplication and division. This can be performed in any order, or they can be solved together.
= 60 + 9 – 6 [Perform multiplication and division]
= 63 [Perform addition and subtraction]
Why do we follow the PEMDAS Rule?
As we have discussed, a math operation consisting of multiple mathematical operations can be solved in more than one way. In that case, if a teacher asks 5 students to evaluate an expression, each student can come up with a different answer. But there can only be one correct answer, right? The correct answer can only be obtained by following the PEMDAS rule. Hence, it is essential that we stick to the PEMDAS rule while solving a math expression with multiple operations.
For example, we can solve the previous expression without solving the PEMDAS rule. Let us find out what happens if we neglect the rule.
(4 + 8) x 5 + 9 – 12 \(\div ~\sqrt{4}\)
= 12 x 14 – 12\(\div ~\sqrt{4}\) [Perform addition]
= 12 x 2 \(\div ~\sqrt{4}\) [Perform subtraction]
= 24 \(\div ~\sqrt{4}\) [Perform multiplication]
= 24 \(\div\) 2 [Evaluate exponents]
= 12 [Perform division]
Originally, the value of the expression was 63. But the value of the expression became 12 when we neglected the PEMDAS rule. So, it is clear that we need to follow the rule to arrive at the correct answer.
Solved Examples
Example 1: Evaluate (8 – 2) x 8 \(\div~ 2^2\) + (3 – 1).
Solution:
According to the PEMDAS rule, the sequence of evaluation should be parentheses, exponents, multiplication, division, addition, and subtraction.
(8 – 2) x 8 \(\div~ 2^2\) + (3 – 1)
= 6 x 8 \(\div~ 2^2\) +2 [Simplify terms inside the parentheses]
= 6 x 8 \(\div\) 4 + 2 [Simplify the exponents]
= 48 \(\div\) 4 + 2 [Perform multiplication]
= 12 + 2 [Perform division]
= 14 [Perform addition]
Example 2: Evaluate (1 + 2)\(^2\) x 5 \(\div\) 3 – 2.
Solution:
We will use the PEMDAS rule to find the value of the given expression.
(1 + 2)\(^2\) x 5 \(\div\) 3 – 2
= 3\(^2\) x 5 \(\div\) 3 – 2 [Simplify terms inside the parentheses]
= 9 x 5 \(\div\) 3 – 2 [Simplify the exponents]
= 45 \(\div\) 3 – 2 [Perform multiplication]
= 15 – 2 [Perform division]
= 13 [Perform subtraction]
Example 3: A football coach ordered cleats for the players in his team. He ordered two pairs of cleats for each of the eleven first-team players and a single pair for each of the four substitute players. Find the total number of football cleats he ordered using the expression 11 x 2 + 1 x 4.
Solution:
The provided expression is 11 x 2 + 1 x 4.
= 22 + 1 x 4 [Perform multiplication]
= 22 + 4 [Perform multiplication]
= 26 [Perform addition]
Therefore the number of football cleats ordered is 26.
You will arrive at the right answer while simplifying an expression only if you follow the PEMDAS rule. Otherwise, the expression may leave different values.
Yes, the PEMDAS rule is known by different acronyms in different countries. PEMDAS is also known as BODMAS (brackets, order, division, multiplication, addition, and subtraction), BEDMAS (brackets, exponents …), and GEMDAS (grouping, exponents…). In all of these cases, the order remains the same. The only difference is in the acronym.
Yes, finding the square root of a number is the same as raising the number to half. Hence, a square root can be considered an exponent.