# Order of Operations

**Order of operations** refers to the conventional order in which mathematical operations must be completed. In general, the rules for order of operations require that we perform operations in the following order:

1) anything in parentheses, then

2) exponents, then

3) multiplication and division, in order from left to right, then

4) addition and subtraction, in order from left to right.

#### Contents

## Common Mnemonics for Order of Operations

There are two common mnemonic devices for remembering the correct order of operations. However, it is important to remember when using these that the true order of operations is defined above, and, in particular, that order of operations makes no distinction between multiplication and division (or addition and subtraction) when ordering.

PEMDAS or "Please Excuse My Dear Aunt Sally":

**P**arentheses**E**xponents**M**ultiplication**D**ivision**A**ddition**S**ubtraction

\[\] BODMAS:

**B**rackets**O**rders (exponents and radicals)**D**ivision**M**ultiplication**A**ddition**S**ubtraction

Note that naively treating either of these can result in errors, as both addition/subtraction and multiplication/division should be grouped and evaluated from left to right. More accurate mnemonics might be PE(MD)(AS) and BO(DM)(AS).

## Addition and Multiplication

One requirement is that we perform multiplication before addition. For example, in the expression \( 2 + 3 \times 4 \), we would obtain the answer \( 5 \times 4 = 20 \) if we begin by adding 2 and 3, but we get \( 2 + 12 = 14 \) if we multiply first. Since it is important that everyone interprets the expression \( 2 + 3 \times 4 \) in the same way, we simply *define* the correct order to be the one in which multiplication is performed first. Thus, \( 2 + 3 \times 4 = 2 + ( 3 \times 4 ) = 2 + 12 = 14 \).

## What is \( 2 \times 4 + 3 \times 5 \)?

Following the correct order of operations, we see that we must evaluate \( 2 \times 4 \) and \( 3 \times 5 \) before we do any addition.

Thus the correct answer is \( 2 \times 4 + 3 \times 5 = 8 + 15 = 23 \). \(_\square\)

## What is \( 48 \div 2 \times 12 ? \)

If we were carelessly following PEMDAS, the answer would be to perform the muliplication first, giving us \( 48 \div 24 = 2 \). However, this is

incorrect.We make no distinction between multiplication and division, and instead complete them from left-to-right. This gives

\[ 48 \div 2 \times 12 = 24 \times 12 = 288. \ _\square\]

## Exponents

We must evaluate any exponents before we add, subtract, multiply or divide. For example, in the expression \( 2^3 + 3 \times 2^2 \), we could obtain a variety of different answers if we changed the order of operations.

If we simply evaluated from left to right, we we obtain the following: \( 2^3 + 3 \times 2^2 \Rightarrow 8 + 3 \times 2^2 \Rightarrow 11 \times 2^2 \Rightarrow 22^2 \Rightarrow 484 \). However, this is incorrect.

The correct ordering requires the evaluation of exponents first, which gives the following: \( 2^3 + 3 \times 2^2 = 8 + 3 \times 4 = 8 + 12 = 20 \), which is now correct.

## What is \( 3^2 \times 2 + 4^3 \)?

Following the correct order of operations, we see that we must evaluate the exponents first. This gives \( 3^2 \times 2 + 4^3 = 9 \times 2 + 64 = 18+64 = 82. \ _\square\)

## Parentheses

In many cases, it is necessary or convenient to manually designate which operations should occur first. Although in general we evaluate 1) exponents, then 2) multiplication and divistion, and finally 3) addition and subtraction, we can designate groups of operations to happen before those by surrounding them with parentheses.

We would normally evaluate \( 2 \times 3 + 4 ^2 \) like this: \( 2 \times 3 + 4 ^2 = 2 \times 3 + 16 = 6 + 16 = 22 \). However, by adding parentheses, we can designate other opreations to occur first: \( \left( 2 \times (3 + 4) \right)^2. \)

Now, starting with the innermost set of parentheses and working out, we have \[ \left( 2 \times (3 + 4) \right)^2 = \left( 2 \times (7) \right)^2 = \left( 14 \right)^2 = 196. \]

## What is \( \left((1+3)^2+4\right)\times 2 \)?

Following the correct order of operations, we have the following: \[ \begin{array}{rlr} \left((1+3)^2+4\right)\times 2 &= \left((4)^2+4\right)\times 2 & \text{ add inside parentheses}\\ &= (16+4)\times 2 & \text{ exponents inside parentheses}\\ &= (20) \times 2 & \text{ addition inside parentheses}\\ &= 40. \ _\square & \text{ multiplication} \end{array} \]

## What is \( \left(3 + 4^2\right)^2 \)?

The parentheses tell us that we must evaluate the expression \( 3 + 4^2 \) first and then square it. We must also be sure to evaluate the \( 4^2\) before adding.

Thus we have \( \left(3 + 4^2\right)^2 = (3+16)^2 = 19^2 = 361. \ _\square\)

## Basic Arithmetic - Word Problems

## Basic Arithmetic - Problem Solving

Vikram has been working with Order of Operations all day, and has a strong sense that

\[ (-a) + b = b - a, \]

but doesn't know why. How many of the \( 11 \times 11 \) ordered pairs of integers \( (a, b) \), where \(a\) and \(b\) are integers from 0 to 10 (inclusive) , are there, such that

\[ (-a) + b = b-a? \]

## See Also

**Cite as:**Order of Operations.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/order-of-operations/