Order of Operations

Introduction

The order of operations is a very simple concept, and is vital to correctly understanding math. Unlike reading, where we always work left-to-right, sometimes with math we need to work one part of a problem before another, or the final answer could be incorrect! We use the term "order of operations" to describe which part of the problem needs to be worked first. Take this equation as an example:

$$ 4+6 \div 2 * 11 = ? $$

If you were to simply solve from left to right, the answer would be incorrect. Let's do that now: 4 + 6 = 10. Divide that by 2 to get 5. Multiply 5 times 11 to get 55. Unfortunately, even though it seemed ok, this answer is wrong.

The correct order of operations

The order of operations will allow you to solve this problem the right way. The order is this: Parenthesis, Exponents, Multiplication and Division, and finally Addition and Subtraction. Always perform the operations inside a parenthesis first, then do exponents. After that, do all the multiplication and division from left to right, and lastly do all the addition and subtraction from left to right.

A popular way of remembering the order is the acronym PEMDAS. Parenthesis, Exponents, Multiply and Divide, Add and Subtract. You can also create a little phrase to go along with this, like "Please Excuse My Dear Aunt Sally." Whatever you choose, make sure that you know all six steps of the order of operations very well.

Let's try solving that equation again, this time using PEMDAS.

$$ 4+6 \div 2 * 11 = ? $$

Step 1) Parenthesis. There aren't any. Move on.

Step 2) Exponents. None. Keep going...

Step 3) Multiplication and Division. Go from left to right performing all the multiplication and division as you come across it, so divide 6 by 2 to get 3, and multiply that by 11 to get 33.

Step 4) Addition and Subtraction. From left to right, 4 + 33 = 37.

$$ 4+6 \div 2 * 11 $$ $$ 4+3*11 $$ $$ 4+33 $$ $$ 37 $$

The whole idea is just to follow the rule: PEMDAS. Now we can try to solve one with parenthesis and exponents.

What if you are given an equation like the one below? Just simplify it in small steps, using the order of operations at all times.

$$ 5+(6*2)^2 \div 5 $$

Remember, the first step is Parenthesis. Look inside the parenthesis, where we have \(6*2\). That comes out to 12. We can then compute \(12^2\) and get 144, making this equation easy to finish:

$$ 5+(6*2)^2 \div 5 $$ $$ 5+(12)^2 \div 5 $$ $$ 5+144 \div 5 $$ $$ 5+28.8 $$ $$ 33.8 $$

By now you should have a basic understanding of the order of operations. To continue learning more about this subject, you can keep browsing our site, or try a web search on Yahoo or Google. MathGoodies.com also offers a great lesson on the Order of Operations.