Order of Operations (PEMDAS)

To quote from my book, Basic Math and Pre-Algebra Workbook for Dummies:

"An arithmetic expression is any string of numbers and operators that can be calculated."

"The word evaluate comes from the word value. When you evaluate an expression, you turn it from a string of mathematical symbols into a single value - that is, you turn it into one number."

It goes on to say that some mathematicians got together and, probably while under the influence, agreed on an order of operations, which is "a set of rules for deciding how to evaluate an arithmetic expression no matter how complex it gets"

So there you have it. No matter how complex and difficult an arithmetic expression gets to be, the order of operations will be there to help you evaluate an expression.

This order of operations sounds pretty important.What is the order?

If you didn't learn PEMDAS at some point in school, then you should file a complaint with the Department of Education

Hopefully this is a review:

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

***Also, and this is very important, always remember to evaluate an expression from LEFT to RIGHT***

Let's start with the first two, addition and subtraction. It's fairly simple to do.

Find -5 + 10 + 2 - 3 - 4 + 6

You can, as I pointed out in my previous post, Negative Numbers and Absolute Value, move any negative numbers to the end of the expression. Personally, I like to "chunk" numbers together. Makes you less likely to make a mistake than if you were to try to solve the expression in one fell swoop.

10 + 2 + 6 - 3 - 4 - 5 = 12 + 3 - 9 = 15 - 9 = 6

Again, if you're just doing multiplication and subtraction, it's not too difficult. Just remember to work from LEFT to RIGHT.

Oh, and keep in mind that regarding PEMDAS, it's more addition/subtraction than solving addition before subtraction. The same goes for multiplication and division.

Find -5 * 10 / 2 * -3 * -4 / 6

Again, I'm going to chunk.

-5 * 10 / 2 * -3 * -4 / 6 = -50 / 2 = -25 * -3 = 75 * - 4 = -300 / 6 = -50

Mixed Operators

 This is where things get complicated. A mixed operator expression contains multiple signs. Working from left to right alone isn't going to cut it. Enter PEMDAS.

 Now we are going to start evaluation multiplication/division from left to right and then addition/subtraction from left to right:

-5 + 10 / 2 -3 * -4 * 4 / 6 = -5 + 5 + 12 * 4 / 6 = -5 + 5 + 48 / 6 = 0 + 8 = 8

Note that I first evaluated expressions based on their operation and then from left to right. If you're trying to practice by solving a worksheet and you find yourself having trouble, get multiple highlighters that are different colors and highlight each operation. It will help, trust me.

A certain physicist I know believes PEMDAS are ridiculous because most expressions will clearly indicate which operations you are to perform first, unless it's a standardized test (where they want YOU to show them that you understand PEMDAS). For example, in the real world, my above expression would be written something like:

(-5) + (10/2) ((-3)(-4)) (4/6)

Anyway, I digress...

Powers aka Exponents

Basically the same stuff as before, separate the operations first and then evaluate from left to right:

1. Exponents
2. Multiplication/Division
3. Addition/Subtraction

-5² + 10/2 -3³ * -4 * 4 / 6 = 25 + 10/2 -27 * -4 *4/6 = 25 + 5 + 108 * 4 /6 = 30 + 432/6 = 
30 + 72 = 102

Parentheses

Arguably, parentheses are the most important factor in the order of operations because once you put parentheses around an operation it gains priority over anything else. Of course, you still have to solve anything in parentheses from left to right:

1. Parentheses
2. Exponents
3. Multiplication/Division
4. Addition/Subtraction

-5² + (10/2) -3³ * (-4*4)/4 = -5² + 5 -3³ * -16/4 = 25 + 5 -27 * - 4 = 25 + 5 + 108 = 30 + 108 = 138

Nested Parentheses

When you have parentheses inside parentheses you evaluate what is inside the inner set of parentheses (or brackets):

{-5² + [10/2 (-3³)] * [(-4*4)/4]} = {25 + [5 -27] * [-16/4]} = {25 -22 * -4} =  25 + 88 = 113

I've found a great site, Math-Aids, for worksheets to practice on. You can generate worksheets based on difficulty and pick what types of operations you would like and it generates worksheets! They offer Order of Operations worksheets too.

For the most part their worksheets do not contain errors but I have seen some before in more difficult problems, such as roots and exponents. But we aren't there yet.