PEMDAS

What is PEMDAS?

PEMDAS is an acronym for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction – a group of words that gives us the order in which to perform the operations in a math expression.

So, if you have two or more operations in an expression, the order of operations – PEMDAS – tells us that we must take care of them in the following order.

Parentheses – $\hspace{0.2em} () \hspace{0.2em}$ or other grouping symbols,
Exponents – powers and roots,
Multiplication and Division (from left to right), and
Addition and Subtraction (from left to right).

Note — As you can see, multiplication and division occupy the same level of hierarchy (#3). One does NOT have precedence over the other. So between them, we solve from left to right, whichever comes first.

The same is true for the pair – addition and subtraction (#4).

Here's a quick example of how we apply PEMDAS when simplifying an expression.

Steps	Comments
$\hspace{1.25em} 14 - {\color{Red} (1 + 5)} / 2 \hspace{0.25em}$	Parentheses before all else
$= \hspace{0.2em} 14 - {\color{Red} 6} \hspace{0.05em} {\color{Teal} / 2} \hspace{0.25em}$	Division before subtraction
$= \hspace{0.2em} 14 - {\color{Teal} 3} \hspace{0.25em}$	Last remaining operation
$= \hspace{0.2em} 11 \hspace{0.25em}$	Answer

If something's not clear at this point, don't worry. We have plenty of examples coming up to help you understand everything.

Why Is PEMDAS Necessary?

Consider the following expression.

2 \times 3 + 4

Depending on what you do first, multiplication or addition, you will get different values for this same expression.

Multiplication before addition

\begin{align*} & 4 + 2 \times 3 \\[1em] = \, \, & 4 + 6 \\[1em] = \, \, & 10 \end{align*}

Addition before multiplication

\begin{align*} & 4 + 2 \times 3 \\[1em] = \, \, & 6 \times 3 \\[1em] = \, \, & 18 \end{align*}

But we don’t want that. We want everyone to arrive at the same result when solving an expression. Otherwise, there will be ambiguity and confusion. And that’s where PEMDAS becomes important.

By specifying the order in which we must solve an expression, it gives an expression a unique value.

As an example, the correct value of the expression above would be 10. Because according to PEMDAS, we must do multiplication (M) before addition (A).

How to Simplify Expressions Using PEMDAS

Example

Simplify : $\hspace{0.2em} 3 \times (7 + 2) \hspace{0.2em}$

Solution

Every time you are simplifying an expression with multiple operations, go over PEMDAS in your head, look for the operations in that order and simplify them.

Let’s do this together for this example.

P is the first letter, so have we got parentheses? Yes, we have. So let’s take care of whatever is within the parentheses first.

3 \times {\color{Red} (7 + 2)} \, = \, 3 \times {\color{Red} 9}

And now we have only one operation to do – multiplication.

3 \times 9 \, = \, 27

And that’s it. We have simplified expressions following the correct order of operations.

Example

Simplify : $\hspace{0.2em} 18 / 3^2 - 2 \hspace{0.2em}$

Solution

Again, we start with P. But there are no parentheses here. So, we move to the next letter – E. We have an exponent, so let’s work on that.

18 / {\color{Red} 3^2} - 2 \, = \, 18 / {\color{Red} 9} - 2

After E, we have MD (remember Multiplication and Division have the same hierarchy). We don’t have multiplication but we do have division. So –

{\color{Red} 18 / 9} - 2 \, = \, {\color{Red} 2} - 2

That leaves us with only one subtraction to do.

2 - 2 \, = \, 0

And that’s our answer.

Example

Simplify : $\hspace{0.2em} 5 + 12 / 3 \times 2 \hspace{0.2em}$

Solution

In this example, we don’t have any Parentheses or Exponents, so we move to MD – and we have both multiplication and division.

Now because the multiply and divide operations have the same hierarchy, we solve them from left to right, as they appear. So, first division and then multiplication.

\begin{align*} 5 + {\color{Red} 12 / 3} \times 2 \, &= \, 5 + {\color{Red} 4} {\color{Teal} \hspace{0.15em} \times \hspace{0.15em} 2} \\[1em] &= \, 5 + {\color{Teal} 8} \end{align*}

And now we just add and get the answer.

5 + 8 \, = \, 13

Done.

Example

Simplify : $\hspace{0.2em} (3 + 5 \times 2) - 1 + 3 + 2^3 \hspace{0.2em}$

Solution

This one looks exciting. Again, first Parentheses.

But this time, inside the parentheses too, we have more than one operation. So we need to apply the PEMDAS rule to that part as well.

That means we will perform multiplication before addition, M precedes A.

\begin{align*} & {\color{Teal} (} 3 + {\color{Red} 5 \times 2} {\color{Teal} )} - 1 + 3 + 2^3 \\[1em] = \, \, & {\color{Teal} ( 3 \hspace{0.2em} + \hspace{0.2em}} {\color{Red} 10} {\color{Teal} )} - 1 + 3 + 2^3 \\[1em] = \, \, & {\color{Teal} 13} - 1 + 3 + 2^3 \end{align*}

Okay, with parentheses out of the way, we move to exponents.

\begin{align*} &13 - 1 + 3 + {\color{Red} 2^3} \\[1em] = \, \, &13 - 1 + 3 + {\color{Red} 8} \end{align*}

Now we are left with a few additions and subtractions. And they have the same priority. So we go left to right.

\begin{align*} & {\color{Red} 13 - 1} + 3 + 8 \\[1em] = \, \, & {\color{Red} 12} {\color{Teal} \hspace{0.2em} + \hspace{0.2em}3} + 8 \\[1em] = \, \, & {\color{Teal} 15} + 8 \\[1em] = \, \, &23 \end{align*}

That’s our answer.

Parentheses Within Parentheses

If you have parentheses within parentheses, you start with the innermost parentheses and move outwards.

Example

Simplify : $\hspace{0.2em} ((4 \times 6 - 15) - 2 + 3^2) / 4 \hspace{0.2em}$

Solution

In this example, we have a pair of parentheses within another pair. So what do we do? We move outwards from the innermost pair. So,

\begin{align*} &( {\color{Teal} (} {\color{Red} 4 \times 6} - 15 {\color{Teal} )} - 2 + 3^2) / 4 \\[1em] = \, \, &( {\color{Teal} (} {\color{Red} 24} {\color{Teal} \hspace{0.2em} - \hspace{0.2em} 15} {\color{Teal} )} - 2 + 3^2) / 4 \\[1em] = \, \, &( {\color{Teal} 9} - 2 + 3^2) / 4 \end{align*}

And now that we have taken care of the nested parentheses, it looks much simpler. We have already solved a similar expression in the last example.

\begin{align*} &(9 - 2 + {\color{Red} 3^2} ) / 4 \\[1em] = \, \, &( {\color{Teal} 9 - 2} + {\color{Red} 9} ) / 4 \\[1em] = \, \, &( {\color{Teal} 7} {\color{Red} \hspace{0.2em}+\hspace{0.2em}9} ) / 4 \\[1em] = \, \, & {\color{Red} 16} / 4 \\[1em] = \, \, &2 \end{align*}

Exponents on Exponents

If you have exponents on exponents, as in the example below, you solve it top-down.

Example

Simplify : $\hspace{0.2em} 3^{2^2} \hspace{0.2em}$

Solution

Alright, we need to work on the top-most exponent first and move down from there.

\begin{align*} 3^ {\color{Red} {2^2}} \, &= \, 3^ {\color{Red} 4} \\[1em] &= \, 81 \end{align*}

Not too bad, right?

Please Excuse My Dear Aunt Sally

PEMDAS is not especially difficult to remember. But if you need some help, you can use the following mnemonic.

Please Excuse My Dear Aunt Sally

Please – Parentheses
Excuse – Exponent
My – Multiplication
Dear – Division
Aunt – Addition
Sally – Subtraction

As you can see, the first letter of each word points to the appropriate math operation in the correct order. Put together, they form PEMDAS.

And that brings us to the end of this tutorial on PEMDAS. Until next time.