Order of Operations

The Order of Operations must be followed to ensure consistent and reproducible results, even when the same problem is solved by different people. For example, person A may look at "6 + 7 × 5" than person B.




Person A may think, "I'll do 6 + 7 first, then multiply it to 5". The results are, however, wrong.

6 + 7 × 5

= 13 × 5

= 65


Person B may think, "I'll multiply 7 by 5 first, then add the result to 6". This results in the correct answer.

6 + 7 × 5

= 6 + 35

= 41


While there are many ways to solve one particular math or algebra problem, the following order of operations must be followed in Arithmetic AND Algebra to get the correct result. The list is from the most important to the least important.

Parenthesis (or brackets)

Exponents/roots

Multiplications/Divisions

Additions/Subtractions


You can remember this as PEMDAS, or Please Excuse My Dear Aunt Sally.

Here are some examples to help you understand how the order of operations list is used. To be sure you've understood all the concepts, make sure that you've at least understood Example 6.

Example 1 (Parenthesis Example)
This is a variation of the earlier example.

Evaluate (6 + 7) × 5.

In this case, be sure to evaluate the bracketed values first, since it's on top of the list. Evaluate them from left to right, i.e. (6 + 4) followed by (5 + 3). After that, evaluate the multiplication operation.

(6 + 4) × (5 + 3)

= 10 × 8

= 80


So the rule here is, parenthesis is king, and evaluate them from left to right.

Example 2 (Nested Parenthesis Example)
In this example, we have the nested parenthesis (7 + 7). Evaluate the most nested parenthesis first (it has no more nested parenthesis in it). Then evaluate the other parentheses from left to right.

(6 + 5) × (3 × (7 + 3))
= (6 + 5) × (3 × 10)
= 11 × 30
= 330

Example 3 (Exponents and Roots)
Typically exponents are grouped together by parenthesis. Note that exponents are at the same level as roots.



Example 4 (Multiplications vs Divisions)
In this case, multiplication and division is at the same level, so it must be evaluated from left to right.

6 × 10 ÷ 2 × 3
= 60 ÷ 2 × 3
= 30 × 3
= 90

Example 5 (Addition vs Subtraction)
Addition and subtraction are at the same level. You can evaluate any pair of addition and subtraction, but it's preferable to systematically evaluate it from left to right.

Method 1 (Systematic, from left to right)
6 + 7 – 3 + 5
= 13 – 3 + 5
= 10 + 5
= 15

Method 2 (Subtraction first, then addition)
6 + 7 – 3 + 5
= 6 + 4 + 5
= 15

Method 3 (Addition first, then subtraction)
6 + 7 – 3 + 5
= 13 + 2 (because -3 + 5 = 2)
= 15

Example 6 (Altogether now)
The following order of operations example may seem complicated, but it's really easy to solve if you approach it systematically. We typically solve for the top and bottom of the fraction separately (also known as the numerator and denominator, respectively). For convenience, in the first step I chose to solve the inner most bracket of the top of the fraction, that is, (-3)4. This simply works out to be (-3)(-3)(-3)(-3) = 81. In the step after that, I add -1 to 81 to get 80.




In the following step, we need to expand -22. A common mistake is to evaluate this as -22 = (-2)(-2) = 4. Take note that -22 = -(22), hence -22 = -4. Subtracting 80 from the result gives us the numerator -84 as a result.




Now we turn our attention to the denominator. Using PEMDAS, we find that multiplication has a higher priority than addition, so 4 × 3 is evaluated first. The rest of the steps are self-explanatory, so I won't be explaining them here.




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