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Instructor:
*Betty Bundly*

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

This lesson will discuss an important acronym for the order of operations in mathematics. Without these guidelines given in PEMDAS, the accurate and objective nature of mathematical calculation could not exist.

**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.

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.

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.

- 24 + 6 / 3 * 5 * 8 - 9

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 + 80 - 9

- 24 + 10 * 8 - 9

- 24 + 2 * 5 * 8 - 9

Lastly, I complete addition (A) and subtraction (S) from left to right.

- 24 + 80 - 9 = 95

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(25) + 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 =
- 36 - 50 + 10

Last, I add and subtract from left to right.

- 36 - 50 + 10 = - 4

If you encounter a calculation with one expression grouped inside another grouping, start with the innermost grouped expression and work your outward, using PEMDAS.

Example Three:

Calculate 6 + 3{72 / 3^2 - 1(2^3 - 1)^2} + 12 / 4. Since there are two grouping symbols, I start calculating the innermost grouped expression first.

- 6 + 3{72 / 3^2 - 1(2^3 - 1)^2} + 12 / 4
- 6 + 3{72 / 3^2 - 1(8 - 1)^2} + 12 / 4
- 6 + 3{72 / 3^2 - 1(7)^2} + 12 / 4

- 6 + 3{72 / 3^2 - 1(8 - 1)^2} + 12 / 4

Now, I use PEMDAS on the next level of grouping.

- 6 + 3{72 / 3^2 - 1(7)^2} + 12 / 4
- 6 + 3{72 / 9 - 1(49)} + 12 / 4
- 6 + 3{8 - 49} + 12 / 4
- 6 + 3{-41} + 12 / 4

- 6 + 3{8 - 49} + 12 / 4

- 6 + 3{72 / 9 - 1(49)} + 12 / 4

Next, I ignore the addition and complete multiplication and division from left to right.

- 6 + 3{-41} + 12 / 4
- 6 - 123 + 3

Last, I complete the addition and subtraction from left to right.

- 6 - 123 + 3
- -117 + 3 = -114

How does PEMDAS apply to a problem such as (15 + 6^2 / 4)/(3 * 7 - 3^2) ? The numerator and denominator of a fraction each act as a separate grouping symbol, so you should simplify the expressions in each separately using PEMDAS. To simplify the expression (15 + 6^2 / 4)/(3 * 7 - 3^2), the steps would be as follows:

- (15 + 36 / 4)/(3 * 7 - 9)
- (15 + 9)/(21 - 9)
- 24 / 12 = 2

- (15 + 9)/(21 - 9)

Any expression in absolute value should also be treated as a grouped expression, using PEMDAS to simplify the expression inside the absolute value symbols. For example, to simplify the expression 20 - | - 32 + (-2)^3 |, the steps would be as follows:

- 20 + |- 32 + (-2)^3 |
- 20 + |- 32 + (-8)|
- 20 + |- 40|
- 20 + 40 = 60

- 20 + |- 40|

- 20 + |- 32 + (-8)|

**PEMDAS** is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. For any expression, all exponents should be simplified first, followed by multiplication and division from left to right and finally addition and subtraction from left to right. The word 'parenthesis' is first in this acronym to indicate that any expression in a grouping symbol, such as parentheses, should be simplified first. This order may also be memorized using the phrase Please Excuse My Dear Aunt Sally.

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