People were so excited about Grant's new product that he started getting requests left and right for his glasses wipers and spray nozzles. So, he started to keep a list of everyone who would want a pair very soon. By the end of 1 day, 36 of his friends had begged him for glasses wipers and 11 of those people each said that they would also like a pair of nozzles to go with their glasses wipers.
Once he realized how many people were excited about his new business, Grant started to wonder how much money he could bring in. So assuming that Grant only sells glasses, wipers, and nozzles, which of these 4 expressions do you think should replace the question mark to make an equation for the total amount of money that he'll earn.
Well, unless Grant plans on paying other people to buy his products, he'll presumably bring in some money each time he sells a nozzle, and each time he sells a set of wipers. This means that the money he earns is going to be equal to the money he makes from his wiper sales plus the money he makes from the nozzle sales, which is just this first choice up here. For example, let's say, just picking numbers out of the air, that Grant decides to sell a wiper set for $5 and one nozzle for $3. Well, then we know that money earned at that point would be equal to $5 plus $3 or let's see, if I was going to pay for these, it would be 5, 6, 7, $8 total. So all we would do to find the total money earned would be to add the cost of each of the products together.
Speaking of specific prices, Grant quickly realized that there was one small issue that he hadn't dealt with yet. He hadn't had a chance to think about how much his products should cost. I randomly picked the numbers $5 for wipers and $3 for a nozzle but he's probably going to want to make a little bit more money than that. For some reason, his friends who had promised to buy his products didn't seem particularly worried about how much they would cost. But he wanted to get closer to crunching some numbers so he could figure out what kind of fortune his Gleaming Glasses products would give him. So right now, we have this very general equation that we came up with in the last quiz for the total amount of money that Grant's going to make, but as we can see there aren't any numbers in this yet. What we want to do is get the actual price of each product to be involved in this equation in some way. So what other information do we already have about this situation? Well, we learned that 36 people want to buy glasses wipers. We also know that 11 people each want two nozzles. What we don't know yet, is the price that Grant should set for each product, a set of wipers and one nozzle. Similar to what we did before, let's pick one variable to represent the cost per wiper set and another variable to represent the cost per nozzle. I think I'll pick w to represent the cost per wiper set and I'll pick n to represent the price per nozzle. Looking back up at this equation we came up with in the last quiz, let's just start by focusing on the amount of money that Grant's going to earn from selling wiper sets. So for now, we're just going to ignore the part of the equation about the nozzles. So, which of these expressions down here do you think should replace this question mark? How much money is Grant going to earn from selling wipers to 36 friends?
So, we decided to call the price per wiper set w. That means that if Grant sells one set of wipers, he'll earn w dollars. If he sells two sets, he'll earn w+w dollars, which is 2w dollars. If he sells three sets, he'll earn w+w+w, or 3w, and so on and so forth. Since you know that he is going to sell 36 wiper sets, that means he's going to make 36w dollars off of the wipers alone. So, 36w is the correct answer.
Now that we've dealt with our money from wipers, let's move on to figure out how much money Grant's going to make, from selling this first batch of nozzles. Remember that, 11 people, each asked him for two nozzles. Also, we decided that n is equal to the price per nozzle. This is for one nozzle, not for two nozzles. So, which of these choices do you think is equal to the money he is going to earn from these nozzle sale so far?
So part of figuring out how to solve word problems well is figuring out what information isn't important for the question that we're trying to answer. We want to know how much money Grant's going to make from selling nozzles. All that this is going to depend on is the price per nozzle and how many people want to buy nozzles. That means that these answers with 36, which has to do with the people who want to buy wipers are irrelevant for this question. Also w is the price per wiper set not per nozzle, so this answer which depends on w can't be correct. And this one even though it also involves an n, involves a w so, it can't be right either. We're only focusing on nozzle sales here. So what are we left with 11, pretty good nothing having to do explicitly with wiper sales. However, these 2 answers 11 and 22 don't involve an n. That will mean that no amount or how many nozles Grant sells, he's going to make either just $11 or $22. I would hope that, for his sake, the more nozzles he sells, the more money he makes so we want to say that those answers aren't correct. So now we have 11 times 2n and 11n. The key here is remembering that each of these eleven people want two nozzles. Not just one nozzle. So each person who buys two nozzles is going to pay him not n dollars but 2n dollars. If 11 people paying 2n dollars so that means we need 11 times 2n which is this answer right here.
So from our last two quizzes, we now have developed a new expression for total money earned by Grant. Instead of money from wipers, we have 36w, where w is the amount that each set of wipers cost. And instead of money earned from nozzles, we have 11 times person pays for one nozzle. Now when I look at this term 11 times 2n, it looks a little bit complicated to me, definitely more complicated than 36w does. Can you think of a simpler way to write 11 times 2n? Now, there are a ton of different ways that we can write 11 times 2n and come up with a mathematically equivalent expression, but I would prefer that we try to come up with a simpler way to write this. So, if you have an idea of how to do that, please type it into this box right here.
So one way to look at the term 11 times 2n is to think that we're adding 2n to itself 11 times. Okay great, we have 11 2ns added together. However we know that 2n is just equal to n+n. So we can write this yet another way. For each 2n we can instead write n+n, so we end up with n+n plus. So, for every 2n that we had in this equation, I wrote n+n instead. Now I actually already counted all these n's up and it turns out that there are 22 of them. So we know that So apparently 11 times 2n is equal to 22n. That means that we can come up to our equation for the amount of money that Grant is going to earn and replace 11 times 2n right here with 22n.
Writing up and counting out those 22 n in the last quiz took me a really long time, so I can only imagine how torturous it would have been if I had, had to multiply there is a much easier way to simplify terms than adding them together like we did then. To talk about how this works, let's start by considering a term. Lets say, 5x times 7y times 2z. Now remember, a term is a bunch of variables or numbers, or both, that are multiplied together. And before we start playing around with this term, I'd like to add one more word to our vocabulary, factor. Now, factors are things that we multiply together. Just like terms are things that we add together. So this term, 5x times 7y times 2z, is made up of a bunch of factors that are all multiplied together. Another way to think of a factor, is that it's something that a term is divisible by. Now, we can't forget that in this term there are invisible multiplication signs between the five and the x, the seven and the y, and the two and the z. So another way to think about this term is that it's
So real quick, let's just have a quiz. Which of these choices down here is a factor of this term, 5x times 7y times 2z? Remember, you can think of a factor as something that this term is divisible by, or as something that you can multiply by some other set of factors to equal this term. Please check as many answers as you think
So, let's just go through these answers 1 by 1 to figure out which ones are correct. Starting off with 2z, we see that this is something that is multiplied by 5x and by 7y in order to equal this term. So, this is definitely a radiance here. Now when we look at 5, we need to remember, again. Invisible multiplication sign between the 5 and the x. If we multiply 5 by x and 7y and get a little bit tricky and if you miss this one Not a big deal. This actually is a factor of this term, and actually, of any term. The reason that 1 is a factor for any term we might have, whether it's a number or a variable, is because if you multiply 1 by that term itself, you will end up with the term that you're looking for. So in our case, the thing that we need to multiply 1 by, in order to get 5x times 7y times 2z is just this term itself. I know that might seem a little bit complicated right now, but you'll have plenty of practice in the upcoming quizzes, to sort this all out. Bottom line, 1 is always a factor. Now at first you might think that 0 is similar to 1 in this way. But 0 is actually not a factor here. We can't divide this expression by 0. That would actually give us a solution that we don't know how to interpret right now. And there's nothing that we can multiply by 0 in order to get this expression, since that y is multiplied by 7. And also by 5X and 2Z to come up with this term. So we should check that one off as well, and X works in the same way so we've checked everything off except for the 0. That was great, I know that factors are alittle bit trickier than terms so I hope that this vocabulary is starting to make a little bit more sense. We'll keep using it really frequently.
So now that we understand what factors were multiplied together to create this term, is there a more simplified way for us to write this? Thankfully the answer is yes. However before we can start to play around with this term, we need to learn about something called the commutative property. Specifically the commutative property of multiplication, since all of our factors are multiplied together here. The commutative property of multiplication basically tells us that it doesn't matter what order you multiply things together in. So, for example we know that 2 times 3 = That means we know that 2 times 3 is just equal to 3 times 2. If you want to talk about this with variables instead of with numbers we could talk about, let's say, x times z times y. Here we have three different variables multiplied together and we can rearrange these factors in any order we want and still get a mathematically equivalant expression. So we could have, a ton of different things maybe z times x times y. That's also equal to y times x times z or z times y times x, and so on and so forth. All these expressions and all the other ones that we could get, by rearranging these in other ways are equal to one another. So remember for multiplication, it doesn't matter what order you multiply things together in. This holds no matter how many things you want to multiply together, and it holds whether the factors are variables or numbers.
Okay, time for a quiz. Now that we've talked about the commutative property of multiplication. Let's see if you can apply it when we're dealing with this term right here. 10 times m times 23 times 7. I would like you to check off all of the expressions down here that are equal to this term. Just a hint, there are probably going to be several that are right answers.
So, according the commutative property, we can rearrange everything that we had multiplied together in this term in whatever order we want, as long as we keep every number we started with, or every factor that we started with. And, we still just multiply those number together. So, first I'm going to check and make sure that every answer choice has a 10, an m, a this answer does it, so I'll put an x next to that. We will not be check, checking this box off. We know that we're only allowed to multiply these factors together, so any answer I see that has an operator besides a multiplication sign in it can't be right. So here, we have some addition. So, This answer isn't right, or this one, or this one. Now, there are two tricky things in here. There are parenthesis in this answer and in this answer. Remember, that another way to indicate multiplication is just to write factors next to one another, all in parentheses. We can do this instead of writing the dots for multiplication. So, in fact, checking these two answers that have the parenthesis in them, knowing now that parenthesis mean multiplication, we see that they are both in fact correct. Looking at our four remaining choices, all of these are also right answers. In fact, there are a number of other right answers that I just didn't chose to write down here. Any order that we create these four factors in as long as use to multiply them together will still be equal to this term.
So going back to this term that we're looking at earlier 5x times 7y times 2z, we can start to use what we just learned about commutativity to rearrange all the factors that are multiplied together here, in whatever way we want to. Now, what I really want in this case is for all the variables to be on the back end of the term and all of the numbers. Are the constant factors to be at the front of the term. So please write in these slots an equivalent version of this term by filling in each slot with either a constant or a variable. So, we're just going to rearrange these terms in an order like this. Think about what answers are allowed because of the commutative property.
Looking at this term, the constant factors are 5, 7, and 2. So, we can write those in the first three slots right here. Because of the commutativity of multiplication it doesn't matter what order we write the 5, 7, and 2 in. Instead, I could have written 2, numbers in these first three slots. For simplicity sake, I'm just going to leave mine written this way, but you're answer is right as long as these three constant factors are in these first three slots. Now, what's left over to deal with are the variable factors. So for those, we have x, y and z. Here, it's important to remember, the invisible multiplication signs between these constant factors in the adjacent variables. Remembering that those are there is what's going to allow us to pull out the variables and move them around. So, I'm just going to keep x, y, and z in the order they're written in and fill them into the slots right here. Now, just like with the constant factors, we could write these variables in any order that we want to. We could write z,y,x, z,x,y, x,y,z, actually, that's what we have, [LAUGH] y,x,z, . Any order these three variables go in is correct as long as they are in these last three slots.
So now, this is what our term looks like, 5 times 7 times 2 times x times y times z. Now, this definitely looks different from how our term originally did, but I wouldn't say that it looks less complicated yet. So our goal for this quiz is to make this look much more simple. So think back to when we were talking about Grant's gleaming glasses last time. We simplified this term, 11 times 2n to equal 22n. By using the same idea that we used to come up with that answer, can you think of a way to simplify this term up here? Try to squish it down as much as you can, and remember to get rid of any multiplication signs that you don't think need to be explicit.
The first thing that I notice when I look at this version of our term is that we actually don't need these dots between the variables, and actually we don't need a multiplication sign between 2 and x either. Inside a term the only place we really need to explicitly indicate multiplication with the multiplication sign is between the constants. Factors, the reason that we really had to write them in between the 5, the 7, and the 2 before, was the dot, this didn't look like we were writing 572 x, y, z. So to start off, I'm just going to get rid of those 3 unnecessary multiplication signs. Remember if you kept them in, your term is still mathematically correct. It just contains a couple of extra symbols that don't necessarily need to be there. So now we have this 5 times 7 times 2 to deal with. Those are just numbers and we know how to multiply numbers together,
So now that we've worked step by step through how to simplify terms, I want you to try simplifying this term on your own. So, in this box to the right, please write the most simplified version that you can find of negative 5s times negative 2r times 3t. Remember, that these parenthesis just indicate multiplication. So that means that negative 2r is being multiplied by negative
Like we did before, we're going to start off by identifying the constant factors in this term, and then moving them to the front of the term. So our constant factors here are negative 5, negative 2, and 3. So we can write negative 5 times negative 2 times 3, as the first three factors of our rewritten version of this term. Then after that come the variables, s, r and t. So you multiply the constant factors by the variable factors. Remember that these multiplication signs between the variables actually aren't necessary. So I'm just going to get rid of those right away. Now, negative 5 times negative 2 is just positive 10. Remember, a negative times a negative. Equals a positive. So then we have 10 times 3, and 10 times 3 is just equal to 30. So we can replace negative 5 times negative 2 times 3 with 30, and then multiply that with the rest of the factors that are left over, srt. So our final simplified version of this term is 30 srt. Now to my eye, this looks much easier to deal with than this initial version of the term did. So I think we are making really good progress toward simplifying different expressions.
So, I've already showed you one convention that we use when we simplify terms. We put the constant factors at the beginning of the term. When we do that, it makes it easy once we simplify because we end up with one coefficient and then all of the variable factors after it. People also tend to put the variable factors in alphabetical order. So, if you wanted to do that, we could say that this also equals 30rst. In order to make the change from 30srt to 30rst, we had to remember the commutative property of multiplication. Since we were allowed to switch the order of r and s without changing the value of this term. So, the order that variable factors are written in is just another convention in algebra. It's just something that's useful for us to use, but it doesn't change the fact that this version of the term that we had initially is mathematically equivalent to either of these other two simplified versions.
Before we get to our final quiz on simplification, it's time that I introduce a new, wonderful tool that's called MathQuill. MathQuill's going to help format our expressions that we write so they look really beautiful when you type them in. That way you can make sure that what you're typing is really what you want to be typing. That the expression that you're writing is what you really want to be entering. This is mostly an issue when it comes to exponents since there's not an automatic exponent key on a keyboard. So, for example, if we want to type x to the 6th, then the keys you need to hit are x, carrot, 6 Now, of course, in order to make a carrot, you might need to type different things depending on what kind of keyboard you have. But you'll notice that when you type the carrot, the cursor in the box will move up into the super script spot, and you'll be able to type up here a little 6. It's pretty cool. Then, of course, if you want to get out of the super script position and back to writing the rest of your expression in the normal spot where you write numbers and variables, you need to type in an arrow key to move back down. So, if you wanted to type x squared y, you'd need to type x carrot 2. But then, to move from being up here to being back down here, you just use the arrow key. And then, you can continue typing. Here are a few things for you to try out. I've already, I've already given you some guidance on what to type for these first two, and then I'd like you to figure it out for these last two. However, it should be pretty easy to check if you're doing it right, because you want what's in the box to just look exactly like what's over here on the left. So, just try these out.
Hopefully, these first two were pretty easy for you because I told you what to type. But for these last two, which probably were a bit more complicated than earlier ones, you can now see exactly which key you should type to get these expressions. If you had trouble with this, just go back and practice it again. We're going to keep using this throughout the entire course, so it's super important that you feel comfortable with this. I know it might feel a little bit funny at first, but I'm sure as you play around with it more, it'll become a lot more natural and intuitive.
Now that we've talked about how to write exponents using MathQuill, I'd also like to tell you what we do with fractions. You'll notice that if you want to type 3 over 4 and get it to look exactly like this, all you need to do is type after a fraction, to get out of being in the denominator of the fraction, you just use a right arrow key. And then, you can continue typing like normal. So, try out those first two with the instructions that I've given you. And then, for the last one, which is a challenge problem, you get to use your new found understanding of our wonderful MathQuill tool to write both exponents and fractions. And even an exponent in a fraction.
I already gave you the keystrokes for the first two. Then, for this last, super complicated one, here they are. Remember that you need to use arrow keys to move out of either being in a superscript position, for exponents, or for moving positions around in fractions. If you didn't play around with this a whole lot, you might want to take a second to go back to the last quiz. Type this one in again, and then use the arrow keys, the different arrow keys, to see where they move you around within the expression. I hope you're as excited about how pretty your answers can look using MathQuill as I am. I think it's really, really awesome.
Let's do one more quiz to practice simplifying terms. This time, please simplify front of the 3, is part of the factor negative 3. We're not subtracting 3x from y. We're multiplying 4x times y by negative 3x. So, keeping that in mind, please write the most simplified version of this term that you can over here.
So as always, we're going to start to simplify this term by identifying the constant factors in it. So for those, we have 4 and -3. And remember, we want to move those to the front of the term, so that they are the first two factors that we have written. So this term is equal to have left over, x, y and x. So right off the bat we know how to deal with two of the factors here. We know that 4 times -3 is just equal to -12. So we can replace 4 times -3 with -12 and then write the 3 remaining factors. There's something interesting here though. We have two variables that are the same. We have x and x. I'm going to rewrite this one more time with the order of the variables switched so that the x's are next to each other. That's just generally a good rule of thumb in algebra is to write things that you think are related to one another next to each other. This x times x is interesting. This is the first chance that we're going to get to use exponents. We've seen them already a few times in this course, but we haven't gotten to write our own factors with exponents in them. So, this may seem like new material or it may be review for you, but x times x can also be written with an exponent as x^2. In the same way, x x x = x^3, and so on, and so forth. Remember that in all of these equations right here, I didn't need to write any of these multiplication signs. If we had just written the x's directly next to one another, without any symbol in between, multiplication would still have been implied. So instead we could just have xx equals x^2. I just wrote this out to be abundantly clear. The exponents show how many of the same factor are multiplied together. Incidentally if 2 x's mulitiplied together is equal to X^2 then when we have just 1x is equal to x^1. This seems a little bit silly to write though, writing X to the 1 is more complicated than writing just x so, we usually don't move from having x to writing x to the first instead. So using this information about exponents, we can replace this x times x with an x^2. And with that information we can rewrite this term as -12x^2y. We're obeying all the conventions we know about how to write terms because we have the constant factor at the front, and then we have our variables in ascending alphabetical order. We have the x^2 before the y. That was awesome, I know that this was a pretty complicated thing that involved a ton of different concepts, some of which we haven't focused on a lot yet. But don't worry if exponents still don't feel totally comfortable, we're going to keep working with them a ton more.
So now that we know how to simplify terms, we can get back to Grant and his gleaming glasses. Earlier, we came up with expressions for the amount of money that Grant's going to make from his friends buying his wipers, and also from them buying his nozzles, and we had an equation for the total amount of money earned. Saying that the total amount of money he's going to earn from his friends is equal to the amount that they spent on his wipers plus the amount that they spent on his nozzles. Now I'm going to add one more equation into the mix for you or rather create one more variable. Total money earned right here takes a long time to write, so instead I'm going to call this a different variable. Let's say a. So, with my other equations up here, I can also write total money earned euals a. So let's combine all this information that we came up with in our earlier quizzes and the new information that I just gave you that a stands for the total amount of money earned. Which of these expressions down here, do you think should replace this question mark, in this equation for a? You can check as many answers as you think are right. I know this is a lot to look at on the screen at one time, but once we come up with this final equation, we'll only have single letters and numbers, and no more words, in our equation, and that will make things much easier for the future.
If we look at the information on the top half of the screen, this equation in orange right here is completely related to everything over here on the left. We know that instead of writing total money earned, we can write a, like we have in this bottom equation. So, we can cross this out and put a. And since we've figured out alternate expressions for money earned from wipers and money earned from nozzles, we can just substitute in those new values in these slots. So, we know that money from wipers is also seen over here, and it equals 36w. So, we can take 36w here, and put it here. We can do the same thing with the money from the nozzles. Money from nozzles is 22n. So, we can put that in this spot. Of course, we want to keep the operators that join these quantities together since we're just calling these quantities different names. So, we still have an equal sign and a plus sign. So, we should have a=36w, move the plus sign down, plus 22n. So our answer choice must be, well here is that exact thing written down, 36w+22n. The question is, are any of these other answers right? Well, we can go through one by one and eliminate ones we know are wrong. This has a bunch of factors multiplied together, so does this one. This one has four different terms added together, and that's not what we're looking for since we only have two terms up here. And all of the rest are at least almost in the form that we're looking for. before. This would mean that 36 people are each paying Grant 2w dollars. We know that in order to buy a set of wipers from Grant, you only pay w dollars, and that 36 people bought wipers, so this answer is not correct. We need something with just answers, there are some incorrect combinations of coefficients and variables. We know that 36 goes with w, not with n, so this answer can't be right, or this one. And we know that the coefficient of n needs to be 22. Since as we figured out a couple of quizzes ago, And we want each of the 11 people who are purchasing nozzles to buy two, so this answer isn't right, either. So, now we're left with just two answers. 36w+2n, which we know is right, and 22n+36w. Now, this expression is actually exactly equivalent to this one. So, it's right as well. In order to say that this is true, we have to use something called the Commutative Property of Addition. This is really similar to the Commutative Property of Multiplication that we've talked about a lot already. Except that this time instead of talking about switching order when you multiply things, we're talking about changing order when you add things together. So, just like we know that That means we know that 2+3=3+2. In the same way, if you have not just numbers but numbers and variables. Let's say, an expression like 36w+2n, you can rearrange the order of the terms that are added together and get an equivalent expression like 22n+36w. So, that means that both of these answer choices were correct.
So now we have this wonderful straight forward equation for the amount of money that Grant's friends are going to pay him once they buy their glasses, wiper, and their spray nozzles. However just when we think we're starting to figure things out for him situation gets a little bit more complicated. Even though this means more work for us, this is great for Grant, because more of his friends have asked if they can buy his products. The day after his first round of requests came in, even more of his friends and his friend's friends heard about his new incredible inventions, and asked him to add to them, to the list of people to buy his products. and 25 more people wanted to buy 2 spray nozzles each. This equation no longer tells us the total amount of money that Grant's going to earn from his friends, and his friend's friends buying his wipers and nozzles. So, what we need to do is create a new equation. We need to come up with a different way to express a that takes into account this new information that we have, but also doesn't forget what we had before.
We heard a lot of information in the last video, so here is just a summary of all of that on one screen. We can think about Grant's sales story so far as having two situations. The situation after the first day, which I'm going to call the before picture and the situation after the second day of selling, which is the present situation we're concerned with. The last equation that we came up with, a = 36w + first day. So, I changed the name of the total money earned to a old to show that this is the money earned after the first day of selling. This equation still works if we're only interested in calculating the money that Grant earned from his friends after just that one day, but now, we want to find, is what I'm going to call a new. The money he's going to get from those people and from these new people who also want to buy. What we're going to do in this quiz is come up with an equation for a new, which should combine the information from a old with this new data about people who are also going to buy the wipers and nozzles. So, which of these equations down at the bottom is the correct equation for a new? The total amount of money that Grent will have after these people on the first day buy his wipers and nozzles, and these people from the second day buy his wipers and nozzles. This is a lot of information to take in, so I'm going to lable this quiz a challenge quiz. Actually many of the quizzes that you've had have been challenging, but I think that this one is especially difficult. Think really carefully about how many terms you think this new equation should have. Should it have two terms or should it have four terms? Also, there may be more than one correct answer. Even if you don't get it right the first time, just give it another shot.
So we want a new, what we're looking for, to equal the total amount of money that Grant can count on getting from his friends after the second day of sales, so that needs to account for the money from the people who bought on the first day and the money from the people who bought on the second day. We already know from this equation that we came up with earlier that he'll make 36w+22n Dollars from the initial batch of buyers. So, that means the equation that we have down here still needs to have these terms in it. A couple of the equations don't, so I'm going to cross them off our list. Neither of these answers includes the money from the people bought from Grant on the first day. The other four answers that are left over All have 36W and 22N as terms. So that's great. However this first one only has 36W plus 22N. It doesn't have any information having to do with people who bought on the second day. So that can't be right, since we know that Grant will make some money off of that second round of buyers. To pick between these three remaining equations we need to go through similar steps for people who buy wipers and people who buy nozzles as we did in some of the earlier quizzes to figure out the terms in this equation for a old. We know that the price per wiper is still w. And since 51 more people are going to buy wipers we're going to need to add 51w in. In our equation. This choice doesn't have 51 w. It has 50 w. So it can't be right. Then for the money he'll earn from these 25 extra people buying two nozzles each is going to be = 25 2 n. Since n is the price per nozzle and they each buy two. If we simplify 25 2 n we actually get 50n. So we also need to have 50n, in our answer. Both of these last 2 choices have correct. The only difference between them, is the order that the terms are arranged in. This is just another case, of the communative property of addition. Awesome job! I know this was a really tough quiz so great job for getting through it.
As I said in the past several videos and quizzes, there are a ton of mathematically equivalent ways of writing any expression we might have. For example, if I rearrange the factors within any of the terms here, or rearrange the order of the terms themselves, or let's say, tack on a 0 at the end. The value of the expression on the right side of the equation doesn't change. However, there are certain convienient ways of writing expressions. Ways that actually make them simplier to deal with that we're going to want to use. We call this process of rewriting expressions; simplifing expressions. One really useful tool that can help us start to simplify and expression, or see if we can simplify it any further, is to change the order of the terms. We know that we're allowed to do this because of the communitive property of addition. So, how could we do that here? Let's just rearrange the terms on the right side of this equation so that both terms containing w are in the first two slots and the two terms containing n are in the last two slots. So please just type the proper terms into the proper slots. These smaller spaces are for plus or minus signs but you can also type in as you see fit.
So hopefully, this quiz is a little bit easier than the last one we did. What we're trying to do first, is just identify the terms that have w in them. So, going through our equation up here, we see 36w and 51w. And we can just fill those in to these first two slots. Remember that it doesn't matter which order we write them in. It doesn't matter if we write 36w or 51w first, because of the commutative property of addition. Next we need to identify the n terms, so those are the two that are left, 22n and 50n. And we just want to write those in these two slots. Again, order does not matter because we're just going to add everything together. I'm just going to put the 22n first, since 22 happens to be my favorite number. The last step to make this a fully blown equation, is to add in our plus or minus signs between the terms. Here, in the top equation, every term is positive. We don't see any negative signs here. So that means that each of these signs is going to be a plus sign. When we switch terms around, there's no change to the value of each term. Nothing is going to become negative that used to be positive. So the only thing that's different about this bottom equation from this top equation is the order that we've written the terms in.
So now that we have this equation for the total amount of money that Grant's going to make from people who bought his products the first day and the second day, it would be great if we could simplify it further, however we don't know how to do that yet. So, before we dive into this particular example, let's look at a slightly less complicated but similar problem. Let's say we just have some expression. I picked 3x - 5y + 6x + 9y. Now how can we simplify this expression? Well, I think we need to remember what each of these terms really means. So 3x for example is actually equal to x + x + x and 6x is actually equal to 6 x's added together. Right now, writing this out might not seem particularly helpful. But I think the reason that I chose to write these terms this way will becomes clear if we just rearrange the order of the terms in this expression. So now we switch the order of the terms around so that terms with x are next to each other and the terms of y are next to each other. Now we can see that the first thing we want to do is add 3x and 6x together. With all of the x's written out, that doesn't seem that hard. I just need to figure out how many x's, total I have down here. And if I count them I have 1, 2, 3, 4, 5, 6, 7, 8, that, in order to get the sum of 3x and 6x here We don't actually have to count all the x's. All we really have to do is add the coefficients of the variables together, since 3 + 6 = 9. Once we have the coefficient, we just make sure that we multiply that by the variable that both of these terms have. We usually call this adding like terms, or combining like terms, as I've written at the top of the slide.
Now that we've figured out how to combine figure out how to combine -5y and 9y? So just fill in the term that you think is necessary to make this equation true.
Remember that in order to add 3x and 6x together, I simply added their coefficients and then multiplied by the variable they both contain to get 9x, since 3 plus 6 is 9, and the variable in both terms is x. We can do the same thing to deal with the y terms over here. So negative 5y plus 9y is going to be equal to whatever negative 5 plus 9 is times y. We know that negative 5 plus 9 is equal to that in the box. Over here in our original expression, we have four different terms. And over here, we only have two terms. This is why we call this simplifying expressions. This expression is way more simple than the one over here is. Remember that I was allowed to put an equal sign here, between these two expressions, because they are exactly mathematically equivalent. In order to come up with the expression over here, I only played with the terms inside of this expression. I didn't bring anything in from the outside. The only way I modified it was by squishing together things that were already there.
Since we've now had a little bit of practice with combining like terms, I want to make sure that it's abundantly clear which terms we're allowed to combine and which ones we're not. Terms that you're trying to add together can be combined if and only if two things are true. So first things first, the terms have to have exactly the same variable or variables. So examples of this will be terms like 3xy and 7xy or -4b and 15b. On top of this I have to add that each of these variables that correspond to another have to have the same powers or the same exponents. So, if we have the term 3x^2 and the term 6x, and we try to add those together, we cannot combine them. These are not like terms, even though they have x's in them, this is 3x^2 and this is just 6x to the first power. They don't have the same power, so we can't squish them down into 1 term. Another example of this would be something like 11ab^3 + 12a^2b3. Even if the b^3 factors have the same power, the a here only has a power of 1 and the a here has the power of 2. We cannot add these two things together because one of the variables involved does not have the same power in both terms. So, we cannot squish them down into one term.
Now that we've talked explicitly about what it means to combine like terms, or combine terms that have the same variables with the same powers in the them, I'd like you to try an example of how to do this. Please simplify this expression, Start by figuring out which terms in this expression are like terms, and then figure out how to combine them. Good luck.
Just like we did in the example, we're going to start by rearranging the order of the terms in this expression, so the terms with the same variable are next to one another. Great, now, we have 7x-10x+5y+3y. The next step is to add together the coefficients of the like terms. Great, so now we have -3x, since 7-10 is negative three and the variable there was x, plus make this a little bit prettier, since we have a negative sign at the front, we could because of the commutative property, rearrange the order of the terms one more time. So, I personally prefer to write it this way, 8y-3x. So this is our new version of the expression on the left. It's almost hard to tell from looking at these two expressions that they're actually equal to each other, but that just shows you how powerful combining like terms is. We end up with an expression that's much nicer to look at and much easier to use.
Now, this time we have a expression that's a bit more complicated than the past few we've seen. However, if we keep in mind our rules that we know for combining like terms, that you can only combine terms that have the same variables with the same powers, then this actually shouldn't be that much harder than the other one's we've done. So, just give it a try. Write the simplest version of this expression that you can in this box over here.
The first thing we need to do in order to start simplifying this expression is to identify which terms are like terms. Negative 5x squared is the only term in this expression that has an x to the 2nd power in it. So, it is not like terms of anything else here. However, we have 6x and negative x. At first sight, these might both seem related to the x squared term. But remember, the power of the term matters. Both of these terms have x to the firsts. Remember, there are little invisible ones written when you just have a variable on its own. x is just equal to x to the 1st. Anyway, these two terms are like terms. So, I'm going to start to rewrite this expression with the like terms next to one another. Remember that the sign of the term, it belongs to the term itself, and moves along with it when we rearrange the order. Then the two terms that we haven't written yet are 14 and negative 3. These are just numbers, so they are definitely like terms, and we'll keep them next to one another. Now, that we've identified which terms are like terms, it's time to combine those like terms. So, negative 5x squared stays by itself, since we can't combine it with anything else. For the x to the 1st terms, we end up with a new coefficient of 6 minus 1 or 5, so, plus 5 times the variable, which is x. And then all we have left to deal with are the numbers, 14 minus 3 is 11 so we add 11 to the end. Once we've written down our final expression, we can check and make sure that none of the terms in it are like terms of one another. That is, we only have one term of each type in the final expression. This is an x squared term, this is an x to the 1st term, and this is a constant term. So, there's nothing more that we can combine. That means that we've reached the final stage of our simplification. Awesome.
By now you've had a lot of practice with simplifying expressions. So here's a sort of challenge problem for you. Please try to simplify x^2 + 3xy - 7yx - y^2. As always, type your answer into the box right here. Now this is definitely more difficult, or at least a little bit trickier than what I've asked you to do before, but just give it a try.
So, like we've done a couple of times before, let's start off by going through the terms one by one to see if any of them are like terms. X squared right here, is the only terms with an x squared in it. And negative y squared over here is also the only term with a y squared in it. However we have these two middle terms that each have an x to the first and a y to the first. In them. So, are they like terms. Well, according to our rules up here, they do have the same variables, and those two variables, x and y have the same power. The x here is x to the first, as is the x here, and the y here is y to the first, as is the y here. So presumably they are like terms. What we need to do is use the commutative property of multiplication, that we talked about a long time ago, to make these look the same. We know from the commutative property of multiplication that x times y or just xy is equal to y times x or just yx. So I'm going to rewrite this term right here, 7yx as 7xy instead. It's amazing how switching around the orders of factors within terms can change the way an entire expression looks. Now our job is pretty standard. The terms that we're not combining I'm just going to write as they are, so x squared is the same. And then I'm going to combine these two terms in the middle. So remember we add their coefficients. 3 plus negative 7, or 3 minus 7 which is negative 4 times the variables x and y minus y squared which doesn't change
By this point, you've had a bunch of practice with combining like terms. So, let's go back to that problem we were doing with Grant and his glasses wipers and nozzles. Here's the equation we had. a, the total amount of money he's going to earn from his friends buying his products is equal to 36w plus 51w plus 22 n plus 50 n. Last time you saw this, we didn't know how to combine like terms. But, now you do. So, in this box, I would like you to type the most simplified version of this entire equation that you can come up with. Remember, I'm looking for the whole equation, not just this expression. You need to include the a equals in the equation down here.
So, since we know that we are trying to write an equation and not just an expression,I am going to start off by writing a equals in our box down here. So, since I know that the left side of the equation is not going to change at all. Then I can start to combine my like terms. Conveniently, our terms with w are already written next to each other and our terms with n. It was so nice that we did that for ourselves earlier. This is pretty straightforward then. Since these are like terms, I can just add their coefficients. 36 plus 51 is 87. And I multiply that by the variable w. 22 plus 50 is 72. And that gets multiplied by the variable as well. Awesome. We're one step closer to helping Grant figure out how much money he's going to have after these first two batches of friends pay him.
We've simplified terms by combining factors within them, and we've combined like terms. So, let's try to do both of those things at the same time. Here, we have a very long expression, 15x squared plus 3x to the 7th minus 7x times 2x minus That took a really long time to write and a really long time to say. So it would be awesome if we could simplify this. That is what I would like you to try to do. Now if this looks daunting, no big deal, just give it a try. You have absolutely nothing to lose. Writing your work on paper first will definitely help you. Also, remember to try to simplify within each term first, and then once you've done that, to identify what the like terms are in this expression, then you can work on combining them together. Good luck.
The 1st thing that I'm going to, in trying to simplify this expression, is to look at each of the terms and figure out if I can squish it down at all. So first, we have 15x squared, there's nothing that I can do to make that more simple on its own, so I'm just going to rewrite it. Then 3x to the 7th, same thing. Then I come to. Negative 7x times 2x. Now, there are definitely some factors here that I can combine. If I rearrange the order of these things I can put negative 7 and 2 next to each other and then the x's at the end. Negative 7 times 2 is negative 14, so I'll have a coefficient of negative 14. And then we have times x times x and we know that x times x is just. x to the 2nd power. So I can write that as well. Moving on we have negative 4 times 2x. We can again combine the constant factors. So you have minus 8x, negative 5 times 4, 5 times order of those two and then we have 3 times x. Which is just 3x, so plus 3x. Great now each of the terms in and of itself is as simplified as it can get. The next step is to figure out which terms are like terms. I'm also going to already start writing like terms next to each other in a modified version of the equation. We have 2x squared terms, only 1x to the 7th term, 2x to the 1st terms and 1 constant term. Now I just need to combine the terms that I know are like terms. Remember that we add the coefficients of like terms together and then multiply by the variable. 15 minus 14 is 1, so we have 1 times x squared or just x squared, 3x to the 7th is on its own, it's not combined with anything else. Negative 8 plus 3 is negative 5, times the variable x, and then we have our constant term. So this right here is our final answer. Our long, not very pretty expression, has been squished down to be something much more managable.
In the next few minutes, I'm going to throw a couple more vocabulary words into our mix. First, I want to talk about a special kind of expression, in fact, the kind that we've seen most up to this time. These are polynomials, a polynomial is an expression made up of constants, variables or both that are combined using addition, subtraction, or multiplication. So this basically means that we have one or more terms like we've talked about before added together. The variables and I suppose the constants as well in a polynomial, also have to have non-negative integer exponents. So as an example of a polynomial, we might have something like So some polynomials that we see will have more than one variable in them, like these two right here. But often times we'll also see polynomials that have just one variable involved, or maybe even none. So an example of that might be something like But again, any combination of constants and variables, using these operators, and only these kind of exponents makes a polynomial.
So now that we've talked about what polynomials are, and seen a few examples of them, which of these expressions are not polynomials? Notice I want you to check the ones that are not polynomials, not the ones that are polynomials. Remember the three requirements that an expression must fit in order to count as a polynomial. I have listed them right here in case you need to check. And if you have trouble with this just go back and watch the last video to refresh yourself on what apolynomial is. Goodluck.
Three of these are not polynomials, but the rest of them are. We learned that a polynomial is an expression that only uses addition, subtraction, and multiplication to combine constants variables. But this first trace right here has a big division sign in it, dividing one expression by another expression. Now if we had either the numerator or the denominator of this fraction on its own, then both of them would be polynomials. But because they're divided, this is not a polynomial, so we'll check that one off. We'll talk more about expressions that look like this later on the course. One of our other important rules is that a polynomial must have exponents in it that are only non negative integers. Right here we have x to the 1/2 and 1/2 may be positive, but it's not an integer, so that means that this is not a polynomial. We can think about that rule again when we look at this answer choice right here, -6x^5+x^-3-11x+9. All of these terms in here are fine for being part of polynomials, except for this one x^-3, because -3 is a negative integer exponent, not a non-negative integer exponent. So that means this expression is not a polynomial. It may seem a little bit funny that 6 counts as a polynomial. We don't often talk about numbers or variables on their own as polynomials, but they technically do qualify since they do fit our definition. We said that polynomials could have just variables, just constants, or both. So 6 and all other constant terms are technically polynomials. We're going to talk about polynomials throughout the entire rest of the course, so it's great to get a handle on how to find them.
We just said that the degree of return in a polynomial is the sum of all the exponents of all the variables in that term, so the degree is a number. Now knowing that, I would like you to write in each of these boxes, what the degree of the term next to them is. So just fill in the proper number in each of these blanks.
Our first term right here only has one variable in it, x. And the power that x is taken to is 4. So that means that the degree of this term is 4. This is a fourth degree term. Now this next one is a little bit tricky. 100 is just a constant term. It doesn't have any variable factors in it. Or in other words, it has zero variable factors. So this means that it's degree is 0. This is true of any constant term. If we had the number say, 78 here instead, or negative 432 or think of this is that if we desperately wanted to have a variable in a constant term, we didn't want to change the value of the term, what power would that variable need to be taken to? So let's say we have 1/2 here and we want an x in that term. In order to keep this term equal to 1/2 we need to have x be to the 0 power since Any thing, any variable, any constant to the 0 power equals 1. That means we would actually have 1/2 times 1, which is just equal to 1/2. But even if we choose to write the term in this way with a variable involved the power of that variable is 0. So this is still a 0 degree term For this next term y we need to remember that any variable that doesn't have an exponent written explicitly actually has an invisible 1. As its power. So this term has a degree of 1. It's a first degree term. And lastly we have x squared, y squared, z cubed.. So the degree of this term is just going to be 2 plus 2 plus 3, which is equal to 7.
Now that you know what a polynomial is, we're going to talk briefly about a word that we use to characterize a given term in a polynomial. The degree of a term. Now, the degree of a term is equal to the sum of all the exponents of all of the variable that are in that term. So if we have a term like negative 4x cubed, then the degree of this term is 3. Since we have an exponent of 3 for the only variable that's in the term x. Another way we could say this is that this is a degree is 11 here, because for a first variable we see we have a power of 7 and for the second variable we have a power of 4. y and z are the only two variables in this expression, so the degree is equal to the sum of their powers, 7 plus 4 equals 11.
Just as we can talk about the degree of a term, we can also talk about the degree of an entire polynomial. The degree of a polynomial is just equal to the highest degree of any of its terms. So to figure out the degree of a polynomial we first need to figure out the degree of each of its terms just like we did in the last quiz. If we look back at one of the polynomials we used earlier. -12x^7+3x^2+64x, we can see that this is a by first calculating the degree of each term, so this one is a seventh degree term. This is a second degree term, and this is a first degree term, because of the invisible 1 next to this x that stands on its own. The highest number out of 7, 2 and 1 is 7, so this is a 7th degree polynomial.
Considering this definition of the degree of a polynomial that we just discussed, what is the degree of each of these polynomials down here? Please fill in your answer in the box to the right of each expression.
Let's go through these 1 by 1. So first, x cubed plus 6y squared. We want to start by finding the degree of each term. So the degree of this first one is 3 and the second term is 2. Since we have exponents of 3 and 2, respectively. 3 is greater than 2. So this is a third degree polynomial. Next 7 minus 3x plus 8x cubed y. You have degrees of 0, 1, and 4 since we have 3 plus the invisible 1 on this y equals 4. 4 is definitely the largest of these three numbers so this polynomial is of degree 4. So we can do the same thing for the last two polynomials as well. One interesting thing to note is that the degree of a polynomial is usually information that comes from just one term in the expression, the term with the highest degree. But in some cases, like in this last example, might have more than one term that's of the same degree. So for example, here we have x squared which is a second degree term and z squared which is also a second degree term. Having 2 terms of the same degree. Doesn't change what the highest degree we see here is. We just happen to see that highest degree twice. We still follow the typical rule and say that this is a polynomial of degree 2.
So far we've seen polynomials. The terms haven't been written in any particular order. Remember we learned before about the commutative property of addition. Changing the order that you add terms together in has no effect on the value of the expression; a plus b is just equal to b plus a. So any set of terms that you have added together can be written together in any order. And will still be mathematically correct, regardless of which of those orders you choose to use. However, there is a convention that people use in algebra to help them figure out what order to write their terms of a polynomial in. It's pretty simple. The tendency is to write terms from highest degree to lowest degree. So for example, if we have this polynomial, y + 6xy. Minus y cubed. We can first figure out the degree of each term, and then rearrange these terms so that the term with the highest degree comes first. And then the rest of the terms continue in descending order of degree. A polynomial written in this way is said to be written in standard form. So again writing a polynomial in standard form doesn't make it any more mathematically correct than writing the terms in any other order. But mathematicians usually find the standard form a bit more visually appealing and it also can help us understand the polynomial as a whole a bit more quickly than we could if we wrote the terms in a different order. Since in standard form the term with the highest degree comes first, we only need to look at that first term to figure out what the degree of the entire polynomial is. So that's why standard form is particularly convenient.
Please write these two polynomials in standard form. If you need to peek, the definition of standard form is right here in the top right-hand corner, but try to do it without looking up here.
Let's start by finding the degree of each term in either polynomial. So I'll go ahead and do that right now. Once we figure out the degree of each term in either expression, we rearrange those terms that the term with the highest degree comes first and then the rest of the terms go in order of decreasing degree. So for this top polynomial, we'll need the terms to go in order of degree from 3 to 2 to 1 to 0. So that's going to give us 32x cubed minus 16x squared minus 7x plus 12. We can do the same thing for the second polynomial. Written in standard form, this polynomial is negative 12x to the 7th plus 3x squared plus 64x. X. Now that we have them written in this way, we can see right off the bat that this top polynomial is a third degree polynomial and this bottom one is a seventh degree one. So, standard form definitely makes our job easier. Remember that standard form is something that applies to all polynomials, not just ones with single variables or ones with just x's. You'll get more practice with other polynomials in standard form later.
So we've been talking about simplifying terms in the context of simplifying expressions for a while, and our whole goal with doing all this is to make our expressions look prettier. To make them easier to read, but still let us know what kind of math we're talking about. So something we're doing a lot of is taking shortcuts. In fact, most of the time that you do this, you probably don't even realize that, that's what's happening. Lets say for example that we have something like 2 plus 2, plus 2, plus 2, plus 2. So we have five 2's added together. Now normally if we wanted to express this quantity, we wouldn't write it this way. We wouldn't write all the 2's added together. Instead we would say, okay, how many 2's do you want to add? 5, well that's just the same as 5 times actually just a major short cut to repeated addition. In the same way, we can use exponents to indicate repeated multiplication of the same number. So let's say that instead of having 2 plus 2 plus 2 plus 2 plus 2 We have 5 2's multiplied together. In this case, we can use another kind of shortcut. We can rewrite this using an exponent, Just like we talked about in the last quiz. You may have already had a lot practice with exponents in previous courses, but just in case you've forgotten exactly how they work, we're going to talk about that right now. So let's write this, these 5 2's multiplied together, using exponential notation. So we're going to start by identifying the number that we want to multiply it by itself. So in that case this is 2. So let's write that down, and then we count how many of these 2's are multiplied together. So we know that we have 5, And we take that number 5, The number of 2s we're multiplying. And we write that as a superscript of 2. So superscript is just a small number written to the upper right-hand corner of this other number. And we call 2, the number that we're multiplying over and over again, the b ase number, and we this little 5, its exponent. Now, since you're going to be entering most of your answers on a computer for this class, or probably all of your answers, actually, considering this is an online course, you're going to have to type in answers with exponents in a special way. So what you do is you type in the base number, normally, so you would type in a 2. But then, in order to type the number that's in the exponent, or in order to indicate that this 5 is in this exponent slot, we need to write a caret before the 5. So these are the three keys that you will type, in order, to enter this exact answer on a computer. It's like this little arrow here is pointing up to indicate that this number that follows it is going to be in that higher elevated exponent position, as you can see down here. Now if you have more than just a number in the exponent, then you put parentheses around the entire expression that you want to have positioned in that exponent spot.
So basically, exponents give us a convenient notation for showing repeated multiplication of a given number or variable or combination of numbers and variables. So, knowing that, how would you write 7 times 7 times 7 times 7, using exponent notation? So, I know that you can evaluate this expression to equal just a number without any exponent, but for right now, I want you to make sure that you use exponent notation for the question.
So, since 7 is the number that we're multiplying by itself over and over again, it is the base number, and it goes here as the big number that we write first. Now, we have four of these 7's multiplied together so the exponent that we want is 4. So that means that 7 times 7 times 7 times 7 equals 7 to the 4th power. Now, there are several different ways of saying this answer out loud. You can say 7 to the 4th power 7 to the 4, 7 to the power of 4, or just 7 to the 4th. Certain other exponents have special names that we use, but they're pretty self explanatory. For example, 7 times 7 is equal to 7 to the 2nd power. But instead of saying that, we might also say 7 squared, right here. In the same way, if we have three 7's multiplied together, 7 times 7 times 7, we get 7 to the 3rd power. But sometimes we say 7 cubed instead.
This time, we have x times x times x time x times x times x, that's a lot of x's to write down. So, what's a different way that you can rewrite this term using exponent notation? Please just use one number or variable in the base, and one number or variable in the exponent. Remember that on a computer, you need to write the base number and then the carrot sign, and then the exponent number.
x is the thing that we're multiplying by itself. So, it is the base. And then, there are six of them multiplied together, so that is the exponent number. Again, the way that you needed to write this on the computer was x, carrot, 6. This shows that 6 is being shifted up into the exponent slot.
So now I'm going to give you a break, and let you actually write just a number as an answer to a quiz. What number, not written in exponent notation, or anything, is negative 3, that quantity, to the 4th power equal to?
Since we have parentheses around the negative 3 here and the exponent, the 4 is written outside of those parentheses. That means that the exponent applies to the entire quantity that's inside the parentheses. So, the number that counts as the base number is negative 3. Negative 3 is the thing that we want to multiply by itself 4 times. So, that means we can rewrite this. As negative 3 times negative 3 times negative 3 times negative 3. Now you can just evaluate that numerically. Negative 3 times negative 3 is just positive 9. So you get positive
Now, if instead I have negative 3 to the 4th power, what number does that equal? Is this any different from the quiz before this one?
So, let me help you recall the last quiz first things first. Before we show that if we had negative 3, the entire quantity taken the 4th power, that was equal to positive 81. The difference between that question and this question here, is the lack of parenthesis in this problem. This time we need to think about order of operations. I'm not going to do a super in depth review of this here, but we're going to give you plenty of review problems to practice it, and we'll also keep touching on it at different points throughout the course, since it's pretty fundamental to doing algebra or any other kind of math. You may have heard some sort of mnemonic device or acronym earlier to help you remember The order that we do different mathematical operations when we're evaluating expressions. So I, for example, learned PEMDAS. Now, each of these letters stands for a different mathematical operator. The P, here, is for parentheses. The E, here, stands for exponents. The M stands for multiplication. The D stands for division. The A is for addition, and the last but not least, the S is for subtraction. So this can serve as a guideline, to tell us what order we do all these different operations in, when we're looking at some set of numbers and variables combined together with these operators. So in the last question we did, for example, since the P for parentheses comes before the E for exponents, we knew that the negative sign needed to be applied to the three. Since it was inside the parethesis before, before we took that number to an exponent. There are no parenthesis written out here. So we really have, when we have a negative sign in front of a number, remeber is negative 1 times that number. So negative 3 is actually equal to negative 1 times 3. And we take this 3 to the 4th power. The 4 is only attached to this 3 now because there are no parenthesis that surround the negative sign as well. Since PEMDAS tells us that we evaluate exponenents before we multiply we first need to figure out what 3 to 4t h equals before we can multiply that by negative 1. So we know that 3 to the 4th is equal to 3 times itself 4 times. So now I will just multiply all this out and we get negative 81, so you can see that negative 3 to the 4th power gives us a completely different answer than the whole quantity of negative 3 taking to the 4th power. Parenthesis make a huge difference from using exponents so I say please don't forget to take advantage of review exercises we give you. You want to make sure that things like order of operations, that you've probably seen in previous courses, feel completely comfortable for you. So, if you're having trouble at all with this, or just want to make sure that you totally understand it, do a bunch of our practice problems. This is going to help establish a really strong foundation for you as we move forward in this course.
To make sure that you're super solid on order of operations, and evaluating expressions using exponents, here's a kind of fun quiz for you. Please decide whether each of these expressions is equal to 8, negative 8, or something else, neither of those two. Pick the circle and the proper column for each row.
So again, in order to figure out how to evaluate each of these expressions, we need to use our order of operations knowledge. We need to remember our PEMDAS. For this first problem, negative 2 to the third power, we don't have any parentheses, so that means we need to take the exponent into account before the negative sign. So we have negative 1 times 2 to the third power, which is the same as negative 1 times 8. So we get negative 8. The next problem has some parethesis in it, so we need to deal with what's inside of those first. And inside we have negative 2 squared. Just like in this first problem, you need to do the exponent before the negative sign. So what's inside the parethesis here, is negative 4, since we have negative 2 squared, which is negative 4. So this equals 2 times negative 4 which is just negative 8 again. For the third problem, we have an exponent outside of something that is inside of parentheses. SO we know the negative sign is going to be part of the number that is taken to the exponent, so the entire quantity negative 2 is squared. So this is equal to 2 times negative 2 times negative 2. Which is the same as 2 times positive 4, which is positive 8. Again we have a negative sign inside the parenthesis. So it's part of this number that is taken to the 3rd power. So negative 2 times negative 2 times negative 2 is negative 8. Here the negative sign comes outside the parenthesis. So you have negative 1 times positive 2 to the 3rd, 2 to the into the neither category. 2 plus 2 squared, is just equal to 2 plus 4, which is squared plus 2 squared. Since there's not a parenthesis around this negative 2, we know that we need to square 2 before multiplying it by negative 1. So this is actually equal to negative 4 plus 4, and that's equal to 0. Great job, I know this a lot to do in 1 quiz, but I think that having to thi nk through all these different options that are sort of similar to each other could be a really good way to make sure you have your order of operations sense cemented. And if you don't, not a big deal at all. Remember, just go back and practice a little bit more.
Just to change things up a little bit, what if we have 3^2 times 3^4? That's going to equal three to some power, but what is the exponent that should go here?
Let's start by expanding out each of the things that we have with an exponent. So, we know that 3^2 is just equal to 2 threes multiplied together, so 3 times 3. And we know that these two things are also multiplied together, so we need a multiplication sign between these two threes and these four threes. So, again, we just have a bunch of threes multiplied together and, as we know, exponents are just a tool for writing repeated multiplication of the same number in a shorthand way. So our base is going to be three since three is being multiplied over and over again, and we have six of them multiplied together if you count them out. So the exponent is six. Remember, typed in, you need to type 3^6.
So, what we saw in this last quiz was really interesting. Remember that we started out with 3^2 times 3^4, and we ended up with 3^6 power. What we need to notice here is how the exponents are related to each other. 2+4, our two original exponents, is equal to 6, the final exponent. So, this actually shows us a general rule for multiplying factors that have exponents. If the two numbers are multiplying together have the same base, then their exponents just add to one another, and the base stays the same. So, let's write that in a more general way. If we have some number, or variable a, and it's taken to the power of b, and that's multiplied by another number a that's taken to the power c. Then together, that multiplication can be written as a^b+c. So, like we saw in the last quiz, in the end. Since exponents indicate repeated multiplication, since each of these factors right here is just the number or variable a being multiplied by itself over and over again some number of times. When they are multiplied together, it's just even more a's multiplied together, or some different number of a's multiplied together. This is just a convenient way for us to not have to write out all of the numbers like we did when we were doing this last quiz.
So, remembering this rule that we talked about before, that the factors that have exponents with the same base are multiplied together, their exponents just add. What would be a different way to write x squared times x to the fourth time times x to the third using exponential notation?
Remember that x squared. It's just x times x or 2 x's multiplied together. X to the 4th is just 4 x's multiplied together and we're also multiplying that by another 3 x's that are multiplied together. So here we have a big string of x's all multiplied together, just like we would expect, considering how many exponents we see here, but the same base of over and over again. So, in total here we have 9 x's all multiplied together. And we know that this just equals x to the 9th power. However, we can or rule to do this in a much shorter and easier way. We can get the same answer by just adding these 3 exponents together. We know that our base is still x, but our power is just 2 plus 4 plus
So, what if we have the term x to the 5th times 3 squared times y cubed times x to the 8th? How would you simplify that? Please write your answer in as simplified a form as you can.
Let's first rewrite the factors within this term, so that the factors of the same bases are next to each other. And I'm also going to put the constant factor first. So I'm going to start with 3 squared, then we have 2 factors that have a base of x. So I'll put those next to each other, and then the last thing we have is the y cubed. Well, we know that 3 squared is just 3 times 3, and 3 times 3 is just 9. We also know that when we have x to the 5th times x to the 8th, that's the same as x to the 5 plus 8 power, and nothing is being involved with the y cubed, so it's just going to stay the same. You'll notice I got bit of those multiplication signs between the different factors in the term. Since they are implied, we just don't write them. Now, the thing you need to simplify here is add together the two numbers in the exponent of the x. So you get the final answer of 9x to the 13, y to the 3rd. I think that you're going to find all of this practice with manipulating exponents really useful, as we keep simplifying expressions.
How would you handle this term y^4 x^2 y^-2? Remember, all these are multiplied together. Please write the most simplified version of this term that you can right here.
To solve this problem, we can, once again, use this trick that we learned earlier. We have two factors inside this term that have the same base. So, we have y^4 and y^-2. So, I'm going to start by writing those next to each other. And now, we know that y^4 times y^-2, is just going to be equal to y^4+-2. So, I rewrite the x^2 factor and we have y^4+-2, which is the same as 4- 2power. y^2.
So remember that in the last quiz you simplified y to the fourth x squared y to the negative two to equal x squared y squared. But in doing these we actually dealt with something we haven't talked about explicitly before. This negative exponent right here. My question for you now is basically what do negative exponents mean? So which of these five answers down here. Is an equivalent expression to what we started out, with y to the fourth x squared y to the negative two? Remember whatever you pick down here, also needs to give us the final answer of x squared y squared in the end. So think about what intermediate step you need between here and here. And which one of these answer choices fits that? This is definitely a little bit tricky.
As we saw in the quiz before this one, when we multiplied y^4 times y^-2, we used our short cut to say that this is just equal to, of course, leaving the x^2 there, y^4-2 power. The role that this -2 exponent here plays then, is that it makes it so that, in the final answer, which we know is x^2y^2, we have two fewer y's multiplied together than we did in at least this first part, the y^4 factor of the original expression. But how do you write y^-2 in terms of multiplying some number of y's together. Well if we know that y^4xy^-2=y^2. And also, we remember that y^4 is four ys multiplied together. And similarly, y^2 is two ys multiplied together. This y^ -2 needs to cancel out two of these other four ys that are multiplied together, to give us our final answer. Now, we know that the way that we undo multiplication is division. y^4 needs to be divided by y^2, to equal y squared, we can write this out like this. We have four y's multiplied together, divided by 2 y's multiplied together, and each of these y's in the denominator cancels out one of the y's in the numerator, since everything here is just multiplied together. This is how we end up with just two y's multiplied together. Together, since we can see that we only have y times y left on the left side. If we just look at this way of writing our expression though, we can write both the numerator and the denominator in terms of exponents. So, on the top, we still have y^4 and the bottom of our fraction we have y times y, which is just y^2. So, if we move this way where an expression appear to be end up with y^4 times y^-2=y^4/y^2. We see this factor may be not we so need to multiply this pair x ^ 2 to get a final answer. In this answer choice right here, y^4x^2/y^2.
So, what we learned from the last quiz is that, when you have a negative exponent, you can rewrite whatever term you're dealing with, so that we're instead, dividing by that factor inside the term. So, negative exponents are actually veiled ways of writing division into our term. The way we know what we're dividing by, is to find the factor with a negative exponent, which is just y^-2, in this case. And we move it underneath the factors of the positive exponents to make it the denominator of the term. So, on the right side of the equation where we've moved it, it's now in the denominator. And we've also changed the power of the base to be positive instead of negative. So, we flip the sign of an exponent from negative to positive or from positive to negative. We need to move the factor or the base number that the exponent is applied to, to the opposite side of the fraction. If it's in the top, it needs to move to the bottom and if it's in the bottom, it needs to move to the top, and we also flip the sign of the exponent. So here, we change from negative to positive. So basically, if we have some number, let's call it x and it's taken to a negative power, let's say, that's -a, where a could be any number, or really any expression. Then, this is equal to 1/x^a. So again, we multiply the exponent of a number by -1. We need to switch which part of the fraction the factor it belongs to is part of. I know that what's on the left side here doesn't look like a fraction, but this is actually secretly x^-a/1. So, anything that doesn't look like it's part of a fraction is actually in the numerator of your fraction, where the denominator is 1.
Using what you just learned about signs of exponents and how that relates to the side of the fraction they're on, how could you rewrite 3y-1 x^3 in fraction notation? So, just fill in what you think belongs in the numerator here and what belong in the denominator here. There are a bunch of different mathematically equivalent ways of doing this, but I want you to do it in a way so that you have something that's equal to this expression but so that there are no negative exponents in either the numerator or the denominator over here.
We know that this exponent of -1, attached to the base of y here, means that we can rewrite this factor, y^-1 as 1 / y^1. And we know that y^1 is just y. So y^-1 = of y^-1 in this term. So now, we have All of these factors are just multiplied together, so we can write our fraction right away. In the numerator, we have 3 and x^3. And, the only thing written in the denominator is the y. So, our final answer is 3x^3 / y.
So here's a quiz that looks pretty similar to the last one we did. How can you rewrite 3m ^ 2 / n ^ -4 so that it contains no negative exponents?
What I'm going to do first to deal with this negative exponent factor is separate it out from the rest of the factors. So I'm just going to multiply a little bit more explicitly. So you have 3m^2 times 1/n^-4. However, we know that when we have a negative exponent, n^4 in our case, these factors actually equal to 1/n^4. So I need to replace just the denominator of this fraction right here with this number. So that's going to give us This is really not looking very pretty. However, we know how to handle this. Dividing by a fraction is the same as multiplying by its reciprocal. So, for example, we have 1 over a/b. That's just equal to 1 times b/a. So we can continue modifying this expression using that trick. This is going to equal, keeping the 3m^2, we keep the numerator here, and we multiply by the reciprocal of the denominator. So the reciprocal of 1/n^4 is just n^4/1. Well now this is easy. You know that anything times 1 is just itself. And we know anything divided by 1 is just itself. So this leaves us with 3m^2n^4. So again, we see that switching the sign of an exponent, so switching from negative the n base. Just requires a flipping of the exponent sign and a flipping of the side of the equation that factors on. We switch from having n^-4 in the denominator to having n^4 in the numerator. If you found working with fractions like this a little bit difficult, not a big deal at all. Just take some time to review manipulation of fractions with the materials that we've directed you to.