The PEMDAS Rule (an acronym for “Please Excuse My Dear Aunt Sally”) is a set of rules that prioritize the order of calculations, that is, which operation to perform first. Otherwise, it is possible to get multiple or different answers. We don’t want that to happen.
Below illustrates an example where there are two possible answers. The first solution yields the wrong answer because it calculates the numerical expression from left to right. While the second solution is the correct one because it follows the rules of the Order of Operations.
Simplify the numerical expression.
Wrong Solution:
Correct Solution:
Order of Operations
Step 1: At the top of the list, remember to ALWAYS simplify everything inside the grouping symbols. Examples of grouping symbols are parentheses ( ), brackets , and braces { }. For nested grouping symbols, work it out from the inside and out.
Step 2: Exponential expressions are calculated or evaluated first before performing any of the four fundamental arithmetic operations, namely: addition, subtraction, multiplication, and division.
Step 3: Next, multiply and/or divide whichever comes first from left to right before performing addition and subtraction. This tells us that multiplication and division have a higher level of importance than addition and subtraction.
Step 4: Lastly, add and/or subtract whichever comes first from left to right.
PEMDAS
- PEMDAS is a mnemonic device that can help us remember the order of operations which we already know stands for “Please Excuse My Dear Aunt Sally”.
- P – Parentheses
- E – Exponents
- M – Multiplication
- D – Division
- A – Addition
- S – Subtraction
Just a quick caution, the operations of multiplication and division have the same level of priority.
To decide when to multiply or divide, always perform the one which appears first from left to right.
In the same manner, addition and subtraction are co-equal in terms of importance. Perform the operation that comes first as you work it out from left to right.
Examples of the Applications of PEMDAS Rule
Example 1: Simplify the following expression using the Order of Operations.
Solution
The Order of Operations: PEMDAS
If you are asked to simplify something like “4 + 2×3”, the question that naturally arises is “Which way do I do this? Because there are two options!” I could add first:
4 + 2×3 = (4 + 2)×3 = 6×3 = 18
…or I could multiply first:
4 + 2×3 = 4 + (2×3) = 4 + 6 = 10
Which answer is the right one?
It seems as though the answer depends on which way you look at the problem. But we can't have this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the exact same expression can be calculated so that you can arrive at two or more different answers.
To eliminate this confusion, we have some rules of precedence, established at least as far back as the 1500s, called the “order of operations”. The “operations” are addition, subtraction, multiplication, division, exponentiation, and grouping; the “order” of these operations states which operations take precedence (are taken care of) before which other operations.
A common technique for remembering the order of operations is the abbreviation (or, more properly, the “acronym”) “PEMDAS”, which is turned into the mnemonic phrase “Please Excuse My Dear Aunt Sally”.
This phrase stands for, and helps one remember the order of, “Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction”.
This listing tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and multiplication and division outrank addition and subtraction (which are together on the bottom rank). In other words, the precedence is:
- Parentheses (simplify inside 'em)
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ (3 × 4) = 15 ÷ 12, but is rather (15 ÷ 3) × 4 = 5 × 4, because, going from left to right, you get to the division sign first.
If you're not sure of this, test it in your calculator, which has been programmed with the Order-of-Operations hierarchy. For instance, typesetting the above expression into a graphing calculator, you will get:
Using the above hierarchy, we see that, in the “4 + 2×3” question at the beginning of this article, Choice 2 was the correct answer, because we have to do the multiplication before we do the addition.
(Note: Speakers of British English often instead use the acronym “BODMAS”, rather than “PEMDAS”. BODMAS stands for “Brackets, Orders, Division and Multiplication, and Addition and Subtraction”.
Since “brackets” are the same as parentheses and “orders” are the same as exponents, the two acronyms mean the same thing.
Also, you can see that the “M” and the “D” are reversed in the British-English version; this confirms that multiplication and division are at the same “rank” or “level”.)
The order of operations was settled upon in order to prevent miscommunication, but PEMDAS can generate its own confusion; some students sometimes tend to apply the hierarchy as though all the operations in a problem are on the same “level” (simply going from left to right), but often those operations are not “equal”. Many times it helps to work problems from the inside out, rather than left-to-right, because often some parts of the problem are “deeper down” than other parts. The best way to explain this is to do some examples:
I need to simplify the term with the exponent before trying to add in the 4:
I have to simplify inside the parentheses before I can take the exponent through. Only then can I do the addition of the 4.
4 + (2 + 1)2 = 4 + (3)2 = 4 + 9 = 13
I shouldn't try to do these nested parentheses from left to right; that method is simply too error-prone. Instead, I'll try to work from the inside out. First I'll simplify inside the curvy parentheses, then simplify inside the square brackets, and only then take care of the squaring. After that is done, then I can finally add in the 4:
- 4 + [–1(–2 – 1)]2
- = 4 + [–1(–3)]2
- = 4 + [3]2
- = 4 + 9
- = 13
There is no particular significance in the use of square brackets (the “[” and “]” above) instead of parentheses.
Brackets and curly-braces (the “{” and “}” characters) are used when there are nested parentheses, as an aid to keeping track of which parentheses go with which. The different grouping characters are used for convenience only.
This is similar to what happens in an Excel spreadsheet when you enter a formula using parentheses: each set of parentheses is color-coded, so you can tell the pairs:
I will simplify inside the parentheses first:
So my simplified answer is
The next page has more worked examples examples….
URL: https://www.purplemath.com/modules/orderops.htm
Order of Operations – PEMDAS
“Operations” mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.
But, when you see something like …
7 + (6 × 52 + 3)
… what part should you calculate first? Start at the left and go to the right?
- Or go from right to left?
- Warning: Calculate them in the wrong order, and you can get a wrong answer !
- So, long ago people agreed to follow rules when doing calculations, and they are:
Order of Operations
Do things in Parentheses First
4 × (5 + 3) | = | 4 × 8 | = | 32 | ||
4 × (5 + 3) | = | 20 + 3 | = | 23 | (wrong) |
Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract
5 × 22 | = | 5 × 4 | = | 20 | ||
5 × 22 | = | 102 | = | 100 | (wrong) |
Multiply or Divide before you Add or Subtract
2 + 5 × 3 | = | 2 + 15 | = | 17 | ||
2 + 5 × 3 | = | 7 × 3 | = | 21 | (wrong) |
Otherwise just go left to right
30 ÷ 5 × 3 | = | 6 × 3 | = | 18 | ||
30 ÷ 5 × 3 | = | 30 ÷ 15 | = | 2 | (wrong) |
How Do I Remember It All … ? PEMDAS !
P | Parentheses first |
E | Exponents (ie Powers and Square Roots, etc.) |
MD | Multiplication and Division (left-to-right) |
AS | Addition and Subtraction (left-to-right) |
Divide and Multiply rank equally (and go left to right).
Add and Subtract rank equally (and go left to right)
So do it this way:
After you have done “P” and “E”, just go from left to right doing any “M” or “D” as you find them.
Then go from left to right doing any “A” or “S” as you find them.
You can remember by saying “Please Excuse My Dear Aunt Sally”. | |
Or … | Pudgy Elves May Demand A Snack Popcorn Every Monday Donuts Always Sunday Please Eat Mom's Delicious Apple Strudels People Everywhere Made Decisions About Sums |
Note: in the UK they say BODMAS (Brackets,Orders,Divide,Multiply,Add,Subtract), and in Canada they say BEDMAS (Brackets,Exponents,Divide,Multiply,Add,Subtract). It all means the same thing! It doesn't matter how you remember it, just so long as you get it right.
Examples
Multiplication before Addition:
First 6 × 2 = 12, then 3 + 12 = 15
Parentheses first:
First (3 + 6) = 9, then 9 × 2 = 18
Multiplication and Division rank equally, so just go left to right:
First 12 / 6 = 2, then 2 × 3 = 6, then 6 / 2 = 3
A practical example:
Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?
- Sam uses this special formula that includes the effects of gravity:
- height = velocity × time − (1/2) × 9.8 × time2
- Sam puts in the velocity of 20 meters per second and time of 2 seconds:
- height = 20 × 2 − (1/2) × 9.8 × 22
- Now for the calculations!
Start with:20 × 2 − (1/2) × 9.8 × 22
Parentheses first:20 × 2 − 0.5 × 9.8 × 22
Then Exponents (22=4):20 × 2 − 0.5 × 9.8 × 4
Then the Multiplies:40 − 19.6
Subtract and DONE !20.4
The ball reaches 20.4 meters after 2 seconds
Exponents of Exponents ..
- What about this example?
- 432
- Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:
Start with: | 432 |
32 = 3×3: | 49 |
49 = 4×4×4×4×4×4×4×4×4: | 262144 |
So 432 = 4(32), not (43)2
And finally, what about the example from the beginning?
- Start with:7 + (6 × 52 + 3)
- Parentheses first and then Exponents:7 + (6 × 25 + 3)
- Then Multiply:7 + (150 + 3)
- Then Add:7 + (153)
- Parentheses completed: 7 + 153
- Last operation is an Add:160
Order of Operations Worksheets
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PEMDAS Explained
Please Excuse My Dear Aunt Sally, or PEMDAS, is a way to remember the order of operations in math. Because so much of mathematics depends on getting the order of operations correct, it’s essential to understand the PEMDAS rules inside and out!
But what is PEMDAS? And what did Aunt Sally do that needs to be excused anyway?
Sally knows her order of operations. She has no need for your silly excuses!
Image by LadyBB
The PEMDAS Rules
Let’s talk about what those six letters stand for.
- P is for parentheses (or brackets, or any other grouping symbol).
- E is for exponents (or things such as roots and radical expressions that are equivalent to exponents).
- MD (do multiplications and divisions left-to-right in the same step).
- M is for multiplication.
- D is for division.
- AS (do additions and subtractions left-to-right in the same step).
- A is for addition.
- S is for subtraction.
The PEMDAS rules specify which operations have priority.
- Image by Aha-Soft
- For example, let’s work out 7 + 4 × 52.
- There are no parentheses (P), so first work out the exponent (E).
- 7 + 4 × 25
- Next, you have to multiply (M).
- 7 + 100
- Finally, the only operation left to do is to add (A).
- 107
Ta-Da!!! Not bad, right? Well things can get complicated, so let’s explore some of the tricky cases in detail.
The Left-to-Right Rules
The PEMDAS Rule: Understanding Order of Operations
Everyone who's taken a math class in the US has heard the acronym “PEMDAS” before. But what does it mean exactly? Here, we will explain in detail the PEMDAS meaning and how it's used before giving you some sample PEMDAS problems so you can practice what you've learned.
PEMDAS Meaning: What Does It Stand For?
PEMDAS is an acronym meant to help you remember the order of operations used to solve math problems. It's typically pronounced “pem-dass,” “pem-dozz,” or “pem-doss.”
Here's what each letter in PEMDAS stands for:
- Parentheses
- Exponents
- Multiplication and Division
- Addition and Subtraction
The order of letters shows you the order you must solve different parts of a math problem, with expressions in parentheses coming first and addition and subtraction coming last.
Many students use this mnemonic device to help them remember each letter: Please Excuse My Dear Aunt Sally.
In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS. The BODMAS meaning is the same as the PEMDAS meaning—it just uses a couple different words. In this acronym, the B stands for “brackets” (what we in the US call parentheses) and the O stands for “orders” (or exponents).
Now, how exactly do you use the PEMDAS rule? Let's take a look.
How Do You Use PEMDAS?
PEMDAS is an acronym used to remind people of the order of operations.
This means that you don't just solve math problems from left to right; rather, you solve them in a predetermined order that's given to you via the acronym PEMDAS. In other words, you'll start by simplifying any expressions in parentheses before simplifying any exponents and moving on to multiplication, etc.
But there's more to it than this. Here's exactly what PEMDAS means for solving math problems:
- Parentheses: Anything in parentheses must be simplified first
- Exponents: Anything with an exponent (or square root) must be simplified after everything in parentheses has been simplified
- Multiplication and Division: Once parentheses and exponents have been dealt with, solve any multiplication and division from left to right
- Addition and Subtraction: Once parentheses, exponents, multiplication, and division have been dealt with, solve any addition and subtraction from left to right
If any of these elements are missing (e.g., you have a math problem without exponents), you can simply skip that step and move on to the next one.
Now, let's look at a sample problem to help you understand the PEMDAS rule better:
4 (5 − 3)² − 10 ÷ 5 + 8
You might be tempted to solve this math problem left to right, but that would result in the wrong answer! So, instead, let's use PEMDAS to help us approach it the correct way.
We know that parentheses must be dealt with first. This problem has one set of parentheses: (5 − 3). Simplifying this gives us 2
Order of operations
In mathematics and computer science, order in which operations are performed
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
[1][2] Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20.
With the introduction of exponents in the 16th and 17th centuries, they were given precedence over both addition and multiplication and could be placed only as a superscript to the right of their base.[1] Thus 3 + 52 = 28 and 3 × 52 = 75.
These conventions exist to eliminate ambiguity while allowing notation to be as brief as possible.
Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can indicate an alternative order or reinforce the default order to avoid confusion.
For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, and (3 + 5)2 = 64 forces addition to precede exponentiation. Sometimes, for clarity, especially with nested parentheses, the parentheses are replaced by brackets, as in [2×(3+4)]-5=9.
Definition
The order of operations, which is used throughout mathematics, science, technology and many computer programming languages, is expressed here:[1]
- exponentiation and root extraction
- multiplication and division
- addition and subtraction
This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first.
The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations.
In some contexts, it is helpful to replace a division by multiplication by the reciprocal (multiplicative inverse) and a subtraction by addition of the opposite (additive inverse).
For example, in computer algebra, this allows manipulating fewer binary operations and makes it easier to use commutativity and associativity when simplifying large expressions – for more details, see Computer algebra § Simplification. Thus 3 ÷ 4 = 3 × 1/4; in other words, the quotient of 3 and 4 equals the product of 3 and 1/4.
Also 3 − 4 = 3 + (−4); in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, 1 − 3 + 7 can be thought of as the sum of 1 + (−3) + 7, and the three summands may be added in any order, in all cases giving 5 as the result.
The root symbol √ is traditionally prolongated by a bar (called vinculum) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity.
The parentheses are sometimes omitted if the input is a monomial. Thus, sin 3x = sin(3x), but sin x + y = sin(x) + y, because x + y is not a monomial.
[1] Some calculators and programming languages require parentheses around function inputs, some do not.
Symbols of grouping can be used to override the usual order of operations.[1] Grouped symbols can be treated as a single expression.[1] Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
Examples
1
+
3
+
5
=
4
+
5
=
2
+
5
=
7.
{displaystyle {sqrt {1+3}}+5={sqrt {4}}+5=2+5=7.}
A horizontal fractional line also acts as a symbol of grouping:
1
+
2
3
+
4
+
5
=
3
7
+
5.
{displaystyle {frac {1+2}{3+4}}+5={frac {3}{7}}+5.}
For ease in reading, other grouping symbols, such as curly braces { } or square brackets [ ], are often used along with parentheses ( ). For example:
[
(
1
+
2
)
−
3
]
−
(
4
−
5
)
=
[
3
−
3
]
−
(
−
1
)
=
1.
{displaystyle [(1+2)-3]-(4-5)=[3-3]-(-1)=1.}
Unary minus sign
There are differing conventions concerning the unary operator − (usually read “minus”). In written or printed mathematics, the expression −32 is interpreted to mean 0 − (32) = − 9,[1][3]
Some applications and programming languages, notably Microsoft Excel (and other spreadsheet applications) and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus has higher precedence than exponentiation, so in those languages −32 will be interpreted as (−3)2 = 9.[4] This does not apply to the binary minus operator −; for example in Microsoft Excel while the formulas =-2^2, =-(2)^2 and =0+-2^2 return 4, the formula =0-2^2 and =-(2^2) return −4.
Mixed division and multiplication
Similarly, there can be ambiguity in the use of the slash symbol / in expressions such as 1/2x.[5] If one rewrites this expression as 1 ÷ 2x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
1 ÷ 2 × x = 1 × 1/2 × x = 1/2 × x.
With this interpretation 1 ÷ 2x is equal to (1 ÷ 2)x.[1][6] However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2x equals 1 ÷ (2x), not (1 ÷ 2)x.
For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[7] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[a]
Mnemonics
Mnemonics are often used to help students remember the rules, involving the first letters of words representing various operations. Different mnemonics are in use in different countries.[6][8][9]
- In the United States, the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. PEMDAS is often expanded to the mnemonic “Please Excuse My Dear Aunt Sally“.[5]
- Canada and New Zealand use BEDMAS, standing for Brackets, Exponents, Division/Multiplication, Addition/Subtraction.
- Most common in the UK, Pakistan, India, Bangladesh and Australia[10] and some other English-speaking countries is BODMAS meaning either Brackets, Order, Division/Multiplication, Addition/Subtraction or Brackets, Of/Division/Multiplication, Addition/Subtraction[11][12][13]. Nigeria and some other West African countries also use BODMAS. Similarly in the UK, BIDMAS is also used, standing for Brackets, Indices, Division/Multiplication, Addition/Subtraction.
These mnemonics may be misleading when written this way.[5] For example, misinterpreting any of the above rules to mean “addition first, subtraction afterward” would incorrectly evaluate the expression[5]
10 − 3 + 2.
The correct value is 9 (not 5, as would be the case if you added the 3 and the 2 before subtracting from the 10).
Special cases
Serial exponentiation
If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:[1][14]
abc = a(bc)
which typically is not equal to (ab)c.
However, when using operator notation with a caret (^) or arrow (↑), there is no common standard.[15] For example, Microsoft Excel and computation programming language MATLAB evaluate a^b^c as (ab)c
What is PEMDAS?
PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction.
Given two or more operations in a single expression, the order of the letters in PEMDAS tells you what to calculate first, second, third and so on, until the calculation is complete.
If there are grouping symbols in the expression, PEMDAS tells you to calculate within the grouping symbols first.
The letters PEMDAS and the words parenthesis, exponents, multiplication, division, addition, subtraction may not be very meaningful for someone trying to remember this order, so a phrase has also been attached with the letters in PEMDAS: Please Excuse My Dear Aunt Sally. If you can remember this phrase, then it may be easier to remember the order of operations given in PEMDAS.
Why Is PEMDAS Important?
Without PEMDAS, there are no guidelines to obtain only one correct answer. As a very simple example, to calculate 2 * 4 + 7, I could multiply first, and then add to get 15.
I also have the option to add first, then multiply and get 22.
Which answer is correct? Using PEMDAS, the only correct answer is 15, because the order of the letters in PEMDAS tell me that multiplication, M, should be performed before addition, A.
Here's an explanation of the rules given in PEMDAS:
- P as the first letter means you complete any calculations in grouping symbols first.
- Next, look for exponents, E. Ignore any other operation, and take any numbers with exponents to their respective powers.
- Even though M for multiplication in PEMDAS comes before D for division, these two operations actually have the same priority. Complete only those two operations in the order they occur from left to right.
- Even though A for addition is in PEMDAS before S for subtraction, these two operations also have the same priority. You look for these last two operations from left to right and complete them in that order.
Using PEMDAS in a Mathematical Expression
Example One:
If you are told to calculate or simplify the expression 24 + 6 / 3 * 5 * 2^3 – 9, how does PEMDAS work? First, I look for any grouping symbols (P). There are none, so I then look for any exponents (E). Since I see 2^3, I will do that calculation first, without performing any other calculation.
Now, I look for multiplication (M) and division (D) from left to right, ignoring any addition or subtraction. My next series of calculations will produce the following:
- 24 + 6 / 3 * 5 * 8 – 9
- 24 + 2 * 5 * 8 – 9
- 24 + 10 * 8 – 9
- 24 + 2 * 5 * 8 – 9
Lastly, I complete addition (A) and subtraction (S) from left to right.
Example Two:
Calculate 36 – 2(20 + 12 / 4 * 3 – 2^2) + 10. Since there is a grouping symbol, I must perform all calculations in the parenthesis first, using PEMDAS for any operations in that expression.
- 36 – 2(20 + 12 / 4 * 3 – 2^2) + 10
- 36 – 2(20 + 12 / 4 * 3 – 4) + 10
- 36 – 2(20 + 3 * 3 – 4) + 10
- 36 – 2(20 + 9 – 4) + 10
- 36 – 2(20 + 3 * 3 – 4) + 10
- 36 – 2(20 + 12 / 4 * 3 – 4) + 10
Ignoring the addition and subtraction, I complete the one multiplication operation next.
- 36 – 2(25) + 10
Last, I add and subtract from left to right.
If you encounter a calculation with one expression grouped inside another grouping, start with the innermost grouped expression and work your outward, using PEMDAS.
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