Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
A Mathematical Expression
It might surprise you to find out that we use math on an almost daily basis. Our whole world operates on math; just think about going shopping. What happens when you go to check out? You have your items like the latest video game, a board game, and perhaps some notebooks for school. What happens to these items? They get scanned, and what pops up on the cashier's screen? Why, the price of each item. Isn't this price a mathematical number? Yes, it is. And, what happens as the cashier scans in the rest of your items? They are added to your total cost. At the end, the cashier tells you how much you need to pay to take your items home.
What just happened? Math is what happened. Specifically, addition is what happened. As each item is scanned, it is added to the rest. While you don't see it on screen, there is actually a mathematical expression that is being calculated; that's just an expression with numbers and operations, such as addition, subtraction, multiplication, and/or division.
Say, for instance, the costs of your items are $3.49, $12.99, and $46.99. The mathematical expression that is calculated is 3.49 + 12.99 + 46.99. This, of course, is easy to calculate. But as you get more and more involved in math, you will come across mathematical expressions that also use parentheses and exponents, such as (4 + 5) * (2^2 + 3).
Order of Operations
When you see these kinds of mathematical expressions, you need to be extra aware of the order of operations. Remember that the order of operations tells you in what order to make your calculations. It tells you that you should calculate your parentheses first, then your exponents, then your multiplication and divisions, and then, finally, your addition and subtractions. Only by following your order of operations will you come up with the right answer. If you don't, you will come up with a totally different answer.
For example, if you calculate (4 + 5) * (2^2 + 3) from left to right, ignoring the order of operations, you will get (4 + 5) * (2^2 + 3) = 9 * (2^2 + 3) = 18^2 + 3 = 324 + 3 = 327 for your answer.
Let's see what we get if we follow the order of operations. We'll do the parentheses first. This means that we perform the operations inside the parentheses first. As we perform the operations inside each set of parentheses, we also follow the order of operations. So, inside each set of parentheses, we still will look for parentheses first, then exponents, then multiplication and division, and then, finally, addition and subtraction.
Inside our first set of parentheses, we just have addition, so we can go ahead and perform that. Now, we have 9 * (2^2 + 3). Inside the second set of parentheses, we see an exponent and addition. We perform the exponent before the addition. So, we square the 2. We get 9 * (4 + 3).
Now, we perform the addition inside this set of parentheses. We get 9 * 7. We are now done with our parentheses. Next, are there any exponents? No. What about multiplication and division? Yes, there is multiplication. We get 63.
There isn't any addition or subtraction left, so we're done. Our true answer is 63. Isn't that a big difference from 327? We would have been way off in our answer if we ignored the order of operations.
An easy way to remember the order of operations is to think of the phrase 'Please Excuse My Dear Aunt Sally,' or 'PEMDAS.' The P stands for parentheses, the E stands for exponents, the M stands for multiplication, the D stands for division, the A stands for addition, and the S stands for subtraction. You have all your operations in order.
Remember that you will need to repeat the order of operations inside each set of parentheses. So, inside each set of parentheses, you will need to go through PEMDAS. Once you are finished with each set of parentheses, you can go back to your original PEMDAS and continue on with the E for exponents. Let's look at a couple more examples so you can get a better grasp of working with parentheses and exponents.
7 * (3 + 2)^2
Following the order of operations, we do the parentheses first. We get 7 * 5^2. Now, we do the exponents. We get 7 * 25. Now, we finish with the multiplication. Our answer is 175.
2 + (3^2 * (1 + 2))
Whoa! Look at this problem. We have what is called nested parentheses, where we have one set of parentheses inside another. Let's break this down into the PEMDAS pieces so we don't get confused.
We begin with parentheses. Our first set of parentheses is (3^2 * (1 + 2)). Now, we remember that once inside a pair of parentheses, we have to repeat the order of operations. So, we look for parentheses again. We have (1 + 2). This is 3, so our first set of parentheses is now (3^2 * 3).
Next, we look for exponents inside our first set of parentheses. We have the 3^2. Our first set of parentheses is now (9 * 3). Next, comes multiplication and division. We get 27. We can now replace 27 with our first set of parentheses. We have 2 + 27.
We are now back to our original PEMDAS, where we resume by looking for exponents. We don't have any, nor do we have multiplication or division. So, we finish our problem by performing any remaining addition or subtraction. Our answer is 29.
Let's review what we've learned. A mathematical expression is an expression with numbers and operations, such as addition, subtraction, multiplication, and/or division. When you see parentheses or exponents in such a mathematical expression, you need to be extra aware of following the order of operations.
You can remember your order of operations by memorizing the phrase 'Please Excuse My Dear Aunt Sally,' or 'PEMDAS.' The P stands for parentheses, the E stands for exponents, the M stands for multiplication, the D stands for division, the A stands for addition, and the S stands for subtraction. This tells you to calculate your parentheses first, then your exponents, then any multiplication and division, and then, finally, any addition or subtraction.
Once you are finished with your lesson you should be able to:
- Recall the order of operations using PEMDAS
- Solve a mathematical expression that includes parenthesis and exponents
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