Today, we’re introducing evaluating expressions with order of operations. Let’s jump right in.
Evaluate six plus four times three.
Six plus four times three is a numerical expression. And we need to try and evaluate this expression. We’re actually gonna see how two different students would evaluate this expression. We’ll start with student A who says six plus four equals ten. So that’s the first step. And ten times three equals thirty. Student A has concluded that this expression is equal to thirty. Student B thinks the problem should be solved a little differently. Student B multiplies four times three first. And then add six plus twelve for eighteen.
It seems as though we have a big problem now. Because two students have evaluated the same expression and come up with different answers. This expression, six plus four times three, can only be equal to one of these things. It can’t be thirty and also eighteen. And this is why learning the order of operations is so crucial. In this case, student B was following the order of operations. And student A was not. So let’s just pause this example and take a look at the order of operations. And then we’ll come back to this.
Order of operations are rules that ensure numerical expressions have only one value. Just like in our last example, one student thought it was thirty. And one student thought it was eighteen. But there’s only one correct answer. Expressions have only one value. These rules tell us the order that we should operate or solve expressions. Not following the order of operations is like driving your car the wrong way down a one-way street. So let’s keep our car going the right direction and take a look at what the order of operations are.
We remember the order of operations with the acronym PEMDAS. Or as I like to say “Please Excuse My Dear Aunt Sally.” And what in the world did these letters stand for? Well to start, the P stands for parentheses. So here you see the parenthesis around 𝑥 plus three. During this step, you solve anything that is grouped together, either with parentheses or brackets. The E stands for exponents. During this step, you solve any part of the expression that has an exponent.
The next step is a little bit different. The M and the D stand for multiply and divide. During this step, we will do multiplication and division from left to right. So that doesn’t mean you do all the multiplication first and then all the division. You look at the whole problem and do the multiplication and division from left to right. Our last step is addition and subtraction together. Addition and subtraction function in the same way multiplication and division do. That means we add and subtract from left to right. We don’t solve all the addition and then all the subtraction. We look at the problem as a whole and solve from left to right.
So let’s take these tools and head back to our first example. So “Please Excuse My Dear Aunt Sally” will help us remember how we would evaluate this problem. If we look at the expression and we work our way down the list, we start with P, parentheses. Does this problem have any parentheses? It does not. So we’re done with that step. We’ll also check for exponents. Are there any exponents in this problem? There are not. Now we move on to multiplication and division, remember from left to right. So is there multiplication or division in this problem? And the answer to that is yes. So we must multiply four times three first. That’s our first step here. And you can see that that’s what student B did. And that was the mistake that student A made. They added before they multiplied. Once you multiply four times three, your last step is addition and subtraction. In this problem, that means that we’ll add six to the twelve. Following student B’s example, six plus four times three equals eighteen. Here’s our next example.
Evaluate two minus three plus seven divided by seven.
We’ll need the tools that we learned in order of operations, “Please Excuse My Dear Aunt Sally.” Start this problem by checking the first step in your order of operations, parentheses. Because there are parentheses in this problem, we need to solve what’s inside those parentheses first. In the first step, we subtracted three from two and found that that was negative one. Following the order of operations, we’ll check to see if there are any exponents in our problem. There are not. So we’ll move on to multiplying and dividing from left to right. There was no multiplication. But there was a division. And we need to divide seven by seven. After you divide seven by seven to equal one, there’s only one step remaining in our order of operations. And there is only one step remaining in our expression as well. We just add and subtract from left to right. And in this case, we’ll be adding negative one to one. The solution to this expression turns out to be zero. Here’s another one.
Evaluate four cubed minus nine divided by three squared plus five.
No matter how big the expressions get or how many operations are inside the expression, the steps stay exactly the same. We start with writing down our PEMDAS, “Please Excuse My Dear Aunt Sally,” to help us remember what the order of operations are. Then we copy down the problem and start working our way from the beginning of the order of operations to the end. Are there any parentheses or any other form of grouping in this problem? The answer is yes. We’re gonna solve nine divided by three first. We divided nine by three and then copied the rest of our problem across. So now we have four cubed minus three squared plus five. Our next step, check for exponents. This expression has two sets of exponents. So we need to solve both of those during this step. We have four cubed, which equals 64, and three squared, which equals nine. And that ends all of the exponents in the expression.
Next we’ll check for multiplication and division from left to right. In this case, there is none. Finally, we have addition and subtraction from left to right. But remember that we wanna solve this from left to right. We don’t start with all of the addition. We just start with what’s furthest to the left. In that case, that’s 64 minus nine, which needs to come first. And we’ll solve addition and subtraction from left to right. 64 minus nine equals 55. 55 plus five equals 60. And this expression equals sixty.
To keep your car going the right way, remember “Please Excuse My Dear Aunt Sally.” Parentheses, exponents, multiplication, division, addition, and then subtraction.