The order of operations is a specific order or a set of rules, agreed upon by mathematicians, one must follow when performing arithmetic operations to simplify expressions.
Here is the order for doing operations that you need to follow in the order given below to avoid having different answers when simplifying expressions. If grouping symbols are used such as parentheses, braces, or curly brackets, perform the operations inside the grouping symbols first. Then, proceed with exponents, and so forth...
1. Simplify any expression within parentheses, brackets or grouping symbols: ( ) [ ] { }
2. Simplify powers or expressions involving exponents: 4^{2}, 2^{5}, or 5^{3}
3. Multiply and divide in order from left to right: × and ÷
4. Add and subtract in order from left to right: + and -
Study the example in the figure below carefully so that you understand how to use the order of operations!
Example #1:
4^{2} - 6 × 2 ÷ 4 × 3 + 5
Do exponent:
16 - 6 × 2 ÷ 4 × 3 + 5
Multiply and divide from left to right
16 - 12 ÷ 4 × 3 + 5
16 - 3 × 3 + 5
16 - 9 + 5
Add and subtract from left to right
16 - 9 + 5
7 + 5
12
Example #2:
(2 + 5^{2}) + 4 × 3 - 10
Do parenthesis:
(2 + 25) + 4 × 3 - 10
27 + 4 × 3 - 10
Do multiplication
27 + 12 - 10
Add
39 - 10
Subtract
29
Example #3:
10 - 14 ÷ 2 = 10 - 7 = 3 (Division comes before subtraction)
Remember that if you see multiplication and division at the same time, perform the operation from left to right.
Example #4:
4 + 5 ÷ 5 × 6 = 4 + 1 × 6 = 4 + 6 =10
The following acronyms can make it easier for you to remember the order of operations.
The following mnemonic may help you remember the PEMDAS rule:
PEMDAS (Please Excuse My Dear Aunt Sally)
Even though M comes before D in PEMDAS, the two operations have the same precedence. Same precedence means that multiplication is not more important than division. By the same token, even though A comes before S, the two operations have the same precedence. Addition is not more important than subtraction.
A much better way, in my opinion, to write PEMDAS is P-E-MD-AS.
In P-E-MD-AS, operations with the same precedence have no hyphens between them.
For example, since addition and subtraction have the same precedence, there is no need to put a hyphen between them.
However, P and E have a hyphen between them because P has a higher precedence than E.
All the four letters in MDAS, DMAS, DMAS, and DMAS refer to multiplication, division, addition, and subtraction.
Keep in mind also that PEMDAS, BODMAS, BEDMAS, and BIDMAS are all correct ways to perform the order operations. None of them is better than the other. These are just names that are used, based on the country, to make it easier to remember the rules.
Example #5:
Simplify √4 + 1 + {2 - [(6 - 2) × 5] + 13}.
Work first with the innermost set of parentheses or (6 - 2).
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {2 - [4 × 5] + 13}
Next, work again first with the inner set of parentheses or [4 × 5].
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {2 - 20 + 13}
Stay inside the parentheses until you are done. While working inside the parentheses, notice that you need to add and subtract in order from left to right.
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {-18 + 13}
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + -5
According to BODMAS rule, you need to do root first.
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = 2 + 1 + -5
Add and subtract again in order from left to right
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = 3 + -5
√4 + 1 + {2 + [(6 - 2) × 5] + 13} = -2
The final answer is -2
Dec 03, 22 12:47 PM
Nov 18, 22 08:20 AM