The Rules of PEMDAS
PEMDAS is an acronym that stands for the order of operations that must be carried out when looking at any equation, whether it be one that is complex and long or one that is super simple and relatively easy. When we see an equation that has more than one mathematical symbol (+, -, ÷, etc), we cannot simply read and carry out the equation from left to right. Instead, there are rules we must follow to ensure that the answer we are left with after solving the equation is 100% correct. We must prioritize certain operations, performing them before others. This order can be abbreviated using the acronym PEMDAS!
What does PEMDAS stand for?
PEMDAS stands for parenthesis, exponents, multiplication, division, addition, subtraction. Alternatively, if one has trouble memorizing those terms, one can think of PEMDAS as standing for “Please Excuse My Dear Aunt Sally” in order to easily remember the mnemonic. Either way works, as long as in the end, you think of PEMDAS when solving equations.
How To Use PEMDAS To Solve Equations
The first step one should always do is look for parenthesis (P) in an equation. If an equation has parenthesis, simplify anything within the parenthesis first before doing anything else. If there are multiple operations that are needed to be done within a parenthesis, follow the rest of the order of PEMDAS.
The Incorrect Example: 5(2+3) = ?
One may accidentally first multiply 5 by 2 to get: 10 + 3 = ?
They would then simplify to get the incorrect answer: 13
The Correct Example: 5(2 + 3) = ?
First, solve the expression inside the parenthesis: 5(5) = ?
Second, simplify further to get the correct answer: 25
The second step one needs to do is calculate and/or expand any exponents (E). Whenever there is an exponent in an equation, that needs to be solved after any expressions inside a parenthesis, but before multiplication, division, addition, or subtraction.
The Incorrect Example: = ?
One may only square “3” because they haven’t solved the full parenthesis: 1 + 9 = ?
They would then simplify to get the incorrect answer: 10
The Correct Example: = ?
First, solve the expression inside the parenthesis: = ?
Second, simplify further to get the correct answer: 16
The third step one needs to follow is to carry out any multiplications (M) and/or divisions (D) that occur in the equation you have already simplified after performing steps one and two. Multiplications and divisions have the same level of priority, so if there is more than one of these symbols in an equation, simply just do them from left to right.
The Incorrect Example: 1 + 2 * 3 = ?
One may accidentally first perform the addition before the multiplication: 3 *3 = ?
They would then simplify to get the incorrect answer: 9
The Correct Example: 1 + 2 * 3 = ?
First, perform the multiplication: 1 + 6 = ?
Second, simplify further to get the correct answer: 7
The final step is to add (A) and/or subtract (S) anything that is left in the equation. Just like with multiplication and division, carry out whichever additions or subtractions come first when reading from left to right. After this, the equation should be solved!
The Incorrect Example: 3 * 4 + 6 = ?
One may accidentally first perform the addition before the multiplication: 3 * 10 = ?
They would then simplify to get the incorrect answer: 30
The Correct Example: 3 * 4 + 6 = ?
First, perform the multiplication: 12 + 6 = ?
Second, simplify further to get the correct answer: 18
A Tougher Example
Simplify the following expression: (10 - 24 ÷ 3)2 + 2 * 4 = ?
To solve this, we must first look at what is inside the parenthesis (P). In this case, there is more than one operation inside the parenthesis. Therefore, we must follow PEMDAS. Division (D) comes before subtraction (S) in PEMDAS, therefore we must first divide 24 by 3 to get 8. Then, we subtract 8 from 10 to get a final expression of 2 inside the parenthesis. Our expression has become (2)2 + 2 * 4 = ?
After simplifying the expression inside the parenthesis, we must then look at the exponent (E). In this case, it is squaring the expression inside the exponent. Since we just found that the aforementioned expression is equal to 2, squaring that would give us 4. Our expression has become 4 + 2 * 4 = ?
Next, we see that we are left with just an addition and a multiplication that need to be carried out. Because multiplication (M) comes before addition (A) in PEMDAS, we are first going to multiply 2 by 4 before doing anything else. Our expression has become 4 + 8 = ?
Finally, all we have left to do is add! After doing so, we come to our final answer of 12.
Writer: Reyna Wang
Editor: Rushita Paladugu