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, and 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":
- Parentheses
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
\[\] BODMAS:
- Brackets
- Orders (exponents and radicals)
- Division
- Multiplication
- Addition
- Subtraction
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 division, 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 operations 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 \( \left(3 + 4^2\right)^2 = (3+16)^2 = 19^2 = 361. \ _\square\)