Why does the order matter? Here's an explanation of the order of operations, its importance, some memory devices and some examples with solutions.

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### Why?

The order of operations is the agreement all mathematicians make to solve problems the same way. You know that 4*3+2 = 14, but what is 2+4*3 = ? Shouldn’t it be the same? If we always solve from left to right, the two problems are different. Instead, we solve them in a certain order:

Rule 1: Simplify any operations inside Parentheses

Rule 2: Simplify all Exponents, working left to right

Rule 3: Perform all Multiplication and Division, working from left to right

Rule 4: Perform all Addition and Subtraction, working from left to right

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### Mnemonics

There are several commonly used mnemonics to remember the order of operations. The first is simply PEMDAS, for the first letter of each rule. You can also use: Please Excuse My Dear Aunt Sally. This is helpful since it not only lists the rule in order, “my dear” goes together, just like multiplication and division and “aunt sally” goes together just like addition and subtraction. Sometimes, it helps to write the letters on top of each other, so you can visualize the hierarchy:

- P - Parentheses
- E - Exponents
- MD - Multiplication and Division
- AS - Addition and Subtraction

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### Parentheses

Parentheses are not an operation. Instead, they give you a way to change the order of operations.

Suppose you mow lawns for $12 an hour, but since today is a holiday, you get an extra $3 per hour. If you work for 2 hours, how much money do you make?

Logically, since you get $12/hour + $3/hour that gives 12 + 3 = $15/hour. Then you multiply by 2 hours to get 45.

But using order of operations on the problem 12 + 3 * 2 = ?, the answer is 72, which isn’t correct. Instead, you have to force the addition to happen first using parentheses. So the correct problem looks like this:

(12 + 3) * 2 = 15 * 2 = 45

Now we get the right answer.

Keep in mind that any operations inside parentheses need to follow the order of operations as well.

For example: (3+2*4) * 2

= (3 + 8) * 2

= 11*2

=22

The section 3+2*4 inside the parentheses is solved just like it was a problem on its own.

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### Exponents and Radicals

If you haven’t covered exponents yet, don’t worry about it. You will get to them soon enough. Exponents can be placed on specific numbers or variables, but they can also be placed on parentheses. It works that way because if you do the parenthetical expression first, the exponent always affects a number or variable instead of an expression.

You may be wondering: What about square roots and other radicals? Since radicals are really just fractional exponents (sqrt[4] = 4^(1/2) = 2) they happen at the same time. If it helps, think of the order of operations as: Parentheses, Exponents and Radicals, Multiplication and Division, Addition and Subtraction, but you will rarely see radicals included separately this way.

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### Multiplication and Division--Addition and Subtraction

It’s important to recognize that multiplication and division are on the same level. That means you can do them at the same time. For instance: 9*6/3 can be solved by doing the multiplication first: 9*6 = 54 then the division: 54/3 = 18 or vice versa division first: 6/3 = 2 then multiplication: 9*2 = 18 and you get the same answer. The same rule holds for addition and subtraction.

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### Order of Operations--Homework Help

The only way to get better at using PEMDAS, is to do a lot of problems. You have to train your brain to think in this order. Here are some example problems with solutions. Try to solve each problem before reading through the solution.

The order of operations is useful for word problems, since it helps you write down the correct equations. Use the examples below to learn how to think about problems. While they won’t be the same numbers as your homework, they should cover the same material and help you figure out where you are going wrong.

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### Examples with Solutions

1) 18 + 36 / 3^2

Step 1: Exponents

= 18 + 36 / 9

Step 2: Division

= 18 + 4

Step 3: Addition

= 22

Answer: 22

2) 5^2 * 2^4

Step 1: Exponents

= 25 * 2^4

Step 2: Exponents

= 25*16

Step 3: Multiplication

= 400

Answer: 400

3) 289 – (3 * 5)^2

Step 1: Parentheses

= 289 – 15^2

Step 2: Exponents

= 289 – 225

Step 3: Subtraction

= 64

Answer: 64

4) An interior decorator charges $15 per square foot to lay a carpet, and an installation fee of $150. If the room is square and each side measures 12 feet, how much will it cost to carpet it?

Step 1: If one side of the room is 12 feet, then the area of the room is (12 feet)^2

Problem: 15 * 12^2 + 150

Step 2: Exponents

= 15*144 + 150

Step 3: Multiplication

= 2,160 + 150

Step 3: Addition

= 2, 310

Answer: $ 2,310 to carpet the room.

5) There are 4 hotels in the city that are putting up christmas trees. Each christmas tree uses 7 strands of lights. 2 hotels put up 5 trees and 2 hotels put up 8 trees. How many strands of lights are used?

Step 1: Write the Problem

Problem: 2*(5*7)+2*(8*7)

Step 2: Multiplication in parentheses

= 2*35+2*56

Step 3: Multiplication

=70+112

Step 4: Addition

=182