Order of Operations
Addition, subtraction, multiplication, division, and exponentiation are all operations on the real numbers, meaning things you do to the real numbers. For complicated arithmetic expressions, it's important to perform operations in the correct order. So the "whatever order I feel like" tactic isn't going to work for you all that well.
This correct order is given by the magical phrase "Please Excuse My Dear Aunt Sally" (PEMDAS). Oh, poor, dear Aunt Sally; she gets a little confused sometimes and needs acronyms to remind her how to go about solving her favorite math equations.
The letters stand for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction, in the order we want to do them. Notice that "Multiplication and Division" and "Addition and Subtraction" are grouped together. That's because multiplication doesn't necessarily need to be done before division—you just need to have all your multiplication and division wrapped up before you start in on your addition and subtraction. Don't feel bad if you didn't get that at first. Aunt Sally's been trying to grasp that concept for 40 years and it still eludes her.
First we evaluate things in parentheses. Uh, we can't simplify (4) any more than it already is, so let's move on to the exponents.
Then multiplication and division:
6 – 1
And finally, addition and subtraction:
When adding and subtracting, we work from left to right. Check to see which of your shoes has the big "L" written on the sole of it if you're not sure.
What's 4 – 6 – 2?
We don't have any parentheses, exponents, multiplication, division, or addition, so we jump right away to subtraction. As usual, though, we've gotta move left to right.
4 – 6 – 2 =
(4 – 6) – 2 =
-2 – 2 = -4
Notice that, if we'd subtracted 6 – 2 first, we would get a totally different (and wrong) answer:
4 – (6 – 2) =
4 – 4 = 0
We also work from left to right when evaluating multiplication and division.
What is 3 × 4 ÷ 2 ÷ 6?
We only have multiplication and division here, so let's roll along from left to right.
3 × 4 ÷ 2 ÷ 6 =
12 ÷ 2 ÷ 6 =
6 ÷ 6 = 1
If we worked from right to left, we would get a different answer:
One way to keep track of your work is to break the problem into pieces, separated by addition or subtraction signs. A rock hammer or mortar and pestle should do the trick.
Now work out each of the pieces:
6 + 2 – 0 – 4
Then combine the answers to the pieces:
6 + 2 – 0 – 4 = 4
Yeesh, what a beast. Let's break it down into smaller chunks, each separated by a plus or minus sign (since addition and subtraction come last).
3 + 6 × 16 – 6 × 1
Now we handle that multiplication.
3 + 6 × 16 – 6 × 1 =
3 + 96 – 6
And finally, we rock the addition and subtraction, left to right.
3 + 96 – 6 = 93
Okay, but Please Excuse My Dear Aunt Sally is a really long thing to remember. And we've already spent the last 10 years attempting to block her out. Here at Shmoop, we like to simplify things ("Shmoop" is actually short for "Shmoopalumpagus").
We've seen that subtraction can be replaced by adding a negative, and division can be replaced by multiplying a reciprocal. So all we really need to remember is Please Excuse My Aunt. Do what's inside the Parentheses first, then take all Exponents, then Multiply, then Add. Just remember that division = multiply by reciprocal, and subtraction = add a negative. Yeah, either way, you're going to have to remember some stuff. C'est l'algebra.
Let's work out one last example.
Ok, so let's start by dealing with the stuff inside the parentheses: . We wanna change it into something more manageable before we square it. We should also probably change the radio station. What is that, Avant-Garde Metal?
A pretty roundabout way just to get to the number 1, but we'll take it.
Now that we've finished all operations inside the parentheses, we look for exponents. If they don't immediately present themselves, whistle loudly and shake a bag of treats—they'll come running.
In the first part of the equation we have 12, which of course is just 1. Then we look for multiplication. Since there is none, all that's left is to add -13. The subtraction of such an unlucky number can't be a bad thing.
1 + (-13) = -12
And we're done!