Now that we have a handle on writing numbers in different ways and working with exponents, let's look at how we can calculate answers. Let's start with a story. I set aside a stamp to mail a letter. Then, I made 2 piles of 3 stamps to mail larger packages. How many stamps did I use? Alright, this might be a fairly simple problem, but let's come up with a diagram to model it. This is going to be the first quiz for this section. I'm going to draw some diagrams and I want you to choose which one best represents my story. A square is going to represent a stamp, and a circle around them will represent a pile. Okay, is it diagram A, diagram B, diagram C, or diagram D? Go ahead. Make your choice.
Okay, in my story I had one single stamp, so I notice that in A, B, C, and D. I have a single stamp here. So the difference has to be with the piles. I made 2 piles of 3 stamps, so I know I need to find 2 piles in my diagram. Answer choice B has 3 piles, cause I see the 3 red circles of 2. So this one can't be right. In answer choice C, I have one pile of 6. So, that's not right either. When I look at answer choice A, I have 3 stamps in one pile and has to be correct. D is not correct. I have 4 stamps in 1 pile and 2 stamps in another pile.
So here's my question again, and this is the model that we agreed upon. Let's pair some math with this model. I would have 1 stamp plus 2 piles of 3, or 2 3. Okay, here's what's interesting about this math, I could actually calculate this math two ways. I could do 1 + 2 and get 3 3 which is 9. Or I could do 2 3 which is 6 and then add 1 which is 7. I know that the second answer must be correct. I have 7 stamps in my diagram. So I know that I need to do multiplication first. This 2 has a choice. It can either go with multiplication or it can go with addition. I know that multiplication takes priority. That's so that way, I can get the correct answer of 7. Just like in my model. Okay, so when I think about the order of operations, I know multiplication occurs before addition. Multiplication has to happen before addition, it's at a higher priority level. It turns out that multiplication is also done before subtraction. We know this because subtraction is the same as adding the opposite. For example, 3 - 1 2, subtracting a positive 1 is the same thing as adding a negative 1, or adding the opposite. So I could have 3 + (-1)2. Here the -1 could go with either the multiplication or the addition. I know that multiplication comes first, it has higher priority. So I should do 3 + -2, and I get 1. So when thinking about the order of operations we know addition and subtraction should be on the bottom, they come last. Multiplication should come before addition and subtraction. I'm going to keep track of our order of operations right here. And we'll come back to this later. Alright. Let's check your understanding of this with a quiz.
Okay. I want you to find the value of this expression. Use the order of operations that we discovered. Is your answer 34, 44, make your choice.
When performing the order of operations, you want to make sure we do multiplication first. So I understand multiplication here and here. 2 20 is 40 and 4 3 is 12, now can keep working. I am going to do addition first and then my subtraction. 1 + 40 + 5 makes 46 and then I subtract 12.
So now let's consider division. I know division is the inverse of multiplication, so it should also take higher priority over addition and subtraction. So we know in our hierarchy of the order of operations multiplication and division should come first. Let's consider this problem: -3+24/43. Here are the two expressions that I could write. This first expression has the parentheses around the 24 and the four. The second one has parentheses around the four and the three. In this problem, it's unclear whether or not we're dividing just by four or dividing by 43. We assume that this is true whenever we see the division symbol without parentheses. The number immediately after it is included or divided into the number before it. So there's no need to solve this problem. Here's one other word of caution. Whenever we think about simplifying this expression, we don't actually want to use PEMDAS. If you've heard of PEMDAS then you probably remember that multiplication comes before division, but this is not necessarily true. Multiplication and division have the same priority whenever we do the order of operations. Students who learn that multiplication come first using PEMDAS would be incorrect. We know we should carry out this division before we do the multiplication. In general, we should think about doing our simplification from left to right, because multiplication and division have the same priority in the order of operations. So these parentheses are assumed to be the case, so let's evaluate this expression. I have -3+63 and -3+18. -3+18 makes 15. In doing our order of operations, it's important to remember the hierarchy. That multiplication and division have the same priority and should be done from left to right in the order of operation.
Let's take this question and agree on the number of stamps that we bought using an exponent. Which of these expressions matches my story?
When thinking about an expression that includes an exponent, let's start with one of these squares. I have a 2x2 square. I know I can write 22 as 2^2, but I don't have one square in my book of stamps, I have five, so I need to multipy this by on one book of stamps, but I have two books of stamps, so we multiply this expression by two. If you chose the first one, nice work. This is a pretty tough quiz, so just hang in there and keep trying with these order of operation problems.
Here is the expression that we agreed upon for the last 2 books of stamps. I'm going to show you two ways of doing this and then we're going to figure out which way is correct, based upon our diagram. For this first one I'm going to start in the parenthesis, I know I need to simplify containers first. When I look at this 2 I can choose between the exponent or I can choose multiplication, I'm going to choose the exponent first. So I'll get 2 5 4, I know 2^2 is 4. Right here I have multiplication. Remember that, whenever you have a number outside of a parenthesis, we just have multiplication in front. 5 4 is 20. and I finished simplifying my parenthesis. Now, I can do second one, I'm still going to start in parenthesis. But for this 2, I'm going to choose the multiplication over the exponent. So I'm going to do 5 times 2 first. So I have 2 10^2. I know 10^2 is 100 so I have 2 100 and I get 200. Okay, one of these answers is correct. I know that in 1 book of stamps, I have 20 stamps. So, in 2 books, I must have 40 stamps. So here are two books of stamps. And I can see that I should have method was correct. So let's go back and look. Here's where I made the choice. The two could have been paired with multiplication, or with exponents. I know that the two should be paired with the exponent. The exponent takes priority over, Over multiplication. So with our order of operations we know that we need to add in another priority. Okay, so when face with the decision between an exponent and multiplication I know that the exponent or the power should come first. So powers take the highest priority in our order of operation. When doing my order of operations I know power should come first. Multiplication and division should come second and then addition and subtraction should come third. So you might be wondering where parentheses fit on our order of operations. Well we just need to make sure we do the order of operations within each pa rentheses. We need to simplify any parentheses or brackets before we do any outside operations on them.
Okay, let's try out our order of operations with a quiz. Here's the problem and I want you to tell me its value. You can put your answer in this box here. Remember, if you don't get the question right the first time around, ask yourself, are you making the right choices based on the priority of the order of operations?
All right, when doing the order of operations, I'm going to start within any parentheses I have. I need to simplify this container. So I'm going to leave the I know I have a choice. This 2 could go with multiplication, or with exponents. The power, I know it should go with the power because powers take priority. So I have 42^3 is really 222 or 8-202. Okay, let's keep going with simplifying these parentheses. I have 3 on the outside, now with this 8 I have a choice, it could be paired with multiplication or subtraction. I know multiplication comes first, so I'm going to have 32-202. I do this multiplication before I do the subtraction. My parentheses is getting smaller, which is good. Now the next choice I have to make is with the 20. Should it go with multiplication or with subtraction? I know that multiplication comes first, before addition or subtraction. So I have 3 on the outside, that's -8. So I have 3 on the outside times -8. Alright, if you have trouble with this part here's what you can do. Remember, subtraction is the same thing as adding the opposite, I could have 32+-40. Subtracting positive 40 is the same thing as adding -40. When I think about these 2 numbers I know 40 is more negative. So I'm going to have 8 more negatives than 32. So I get -8, and 3-8 is -24. If you got -24, nice work.
Let's use our order of operations to help us solve this problem. We should solve this problem going from left to right following the hierarchy of the order of operations. Parenthesis and brackets serve as containers and they should be evaluated before we do any operations on to them. Brackets are typically used when we have parenthesis inside of them but we still evaluate them the same way. We start at the very inside and then work our way out. I'm going to simplify this -5^2. I know this 5^2 only goes with the power so I have 25 or -25. Now I simplify this parenthesis and then I add -20 to 3. I lose my brackets because I no longer have parenthesis. I just have -17 so I write that in parenthesis. I can't simplify this parenthesis anymore so now I should go back and do my order of operations. Let's look for powers. I have powers here and here, so I should simplify these numbers. I want to be very careful. I have the negative of 3 ^ 2, and the negative of 2 ^ the 2. So, I have -9, -17, and -4. Adding the opposite is the same thing as subtraction. That's why I can write -17. Finally, I can add all these negative numbers up, and I'll get my final answer. And I get -30. You can think of doing parenthesis and brackets first. We don't include them in our order of operations because, well, they're not operations, they're just containers.
Here's an order of operations problem you can solve, good luck.
Well Chris, that's a really tough order of operations problem. But if you got the answer of -88 great job. The first thing we want to do when we look at a order of operation problem is to do the inner most parenthesis first. I can see right here there is the inner most parenthesis of So the first thing I'm going to do is calculate 6-6^2. doing these order of operation problems, that you write down all the operations that you have not yet performed. So, I'm going to copy down the rest of the problem. So for this first step I simply evaluated what was in the innermost parenthesis, which 6-36 is -30. Now I'm working in the brackets, and I know that I have to do my powers, or my exponents first. So, -5^2 is -25, and 6^2 is 36. Now that I have evaluated all my exponents or powers, the next thing I want to do is my multiplication. When I do my multiplication its a good idea to work from the left to the right, I'm still working in these brackets so the 2 multiplications that I have to do are -25-2, which gives me 50+. I leave the -36, and the other multiplication that I have to do is -1-30, which gives me +30. Now that I have finished all of the exponents, multiplication, and division within my brackets, now I can do my addition and my subtraction, 50-36 is 14, and then I can add the 14 and the 30, to do is this last operation, which is multiplication. -244 gives me -88.
Here's a second problem for the order of operations. Try this one out.
Wow Chris, that's another really hard order of operations problem. If you got -675, you got the answer right. Now this was a tough problem and I even had to use my calculator. Now let's see how I got -675. When I'm looking at this problem, once again I want to work from the inner parenthesis outwards following my order of operations. So the first thing I see is an innermost parenthesis of 1+2. So that's the first operation I'm going to perform. Again, I want to make sure that I copy down all of the other operations that I have not yet performed. Now that I've performed the operation in the innermost parenthesis, I can do the next level of parenthesis which are these brackets. So I want to do -4(3), which is -12. From here I see that I have another operation in parenthesis, which is 4-10. So I can evaluate 4-10, which is -6. Now that I've performed all the operations, inside parentheses, I can move on to my exponents or my powers. So I have three exponents, I have (-6)^2. Remember when the negative is inside the parentheses, it's part of the base, so it's -6-6 which is 36, 3^2 of course is all my exponents I can do my multiplication and division. The only multiplication I have is -5144. So I can write 36+9, -5144 is -720. Now that I've done all my multiplication and division, I can do my addition and subtraction in order from left to right. So first thing I have to do is 36+9 which is 45-720, 45-720 gives me the -675. Great job.