Hello there, children. Today, we are going to work on algebraic expressions and models. Today's educational experience is brought to you by your alumni of algebra, Mr. Faust.
So, let us begin. First thing, just for our own posterity, let's do some sample problems. Without a calculator, evaluate the following.
Now, I know what you're thinking. Negative 2 to the 6th power. Alright. Negative 2 times negative 2. That's 4. times another negative 2, that's negative 8, then it goes to a 16, then a 32, and then a negative, or a positive, oh well, let's see, even exponents, remember, yield us positive results, so it's a positive 64. Now, 3 to the 4th, 3 times 3 times 3 times 3, that's 81. Now, negative 2 to the 6th power.
This is different than this. Why? Because the parentheses are around this, which means this 6th exponent also will affect the negative.
Whereas in this one, the 6th exponent is not affecting that negative. So this is negative 64. Now I know what you're thinking. Will my calculator take care of this? And the answer is it could. but you need to make sure that the problem put something in parentheses you also must put it in parentheses when using a calculator this is the danger of a calculator because numbers one and number three are different so they do need to make note of that ones in parentheses one is not moving on I we have the order of operations first do the operations that in curl within a grouping symbol this could be but not limited to you a parenthesis or brackets you will see both after the grouping symbols taken care of next evaluate the powers that's your exponent or exponents then do multiplication and division from left to right like you're reading a book alright and then last Finally, do additions and subtractions from left to right.
And this becomes PEMDAS. Or, please excuse my dear Aunt Sally. Parentheses, exponents, multiplication and division, and then addition and subtraction. You will also see it as GEMDAS in some cases, because instead of parentheses, they use grouping symbols. Because as mathematics go further and further along, you won't always be limited to a parentheses you may see brackets or some other sort of grouping symbol so take that into account let's move on here now try this harder problem so we have this problem here now let's take a peek my order of operations in PEMDAS parentheses so I do the work inside the parentheses now this is a little parentheses that has no work to do so then I attack this work inside my print see to plus a negative one that is going to give me a one alright now the rest the problem falls down along with it for now what do I do after that exponents pretties experts so wonder the fourth one times Simply just 1. I also have this, 5 squared.
5 squared gives me 25. Alright, now after that, I get multiplication division. That's going to yield me to 6 times 1 being 6, and then subtract after that. Done deal.
Remember, it's important to do things in the order they are supposed to be done. Moving on. Now, we also could have evaluating the following algebraic expressions.
So, they give us an expression here, and then they will give us an x value when x equals 2. So, what exactly do I do? Well, you'll hear this a lot. You can either hear plug and chug, plug and chug, plug and chug, plug and chug, plug and chug, solution, like a little train, quite fun I might add, or you simply will hear substitute and solve, the more higher level of thinking of this.
So what's going to happen is I am going to rewrite the entire problem, except I'm not going to write the letter X. You will also hear this referred to as synthetic substitution. in some higher level math circles, but instead of writing an x, I am simply going to substitute the 2. Now, if this were any other number, I would substitute it.
If it's an x equal 5, I put a 5 in there. x equals negative 12.2, I put negative 12.2 in. Now I go ahead and go to work. I'll write parentheses.
Nothing to do there. Then, after that, in PEMDAS comes exponents. So I take 2 to the 3rd, and I get...
eight i take two squared and i get four now this three exponent does not go to this front number so don't take two squared here and then take three squared there don't take two to the third and two to the third this third power only goes to the stuff in the parentheses really important you also don't wanna multiply first take the stuff in the present see to the third power now we multiply two times 8 and get sixteen three times for and get 12 and then the 27 falls along down there now from here are I've taken care the multiplication now I'm going to go ahead and do my addition 16 plus 12 plus 27 55 done deal moving on now before we get too far ahead of ourselves here we should know some terminology that will be very important for our algebraic lives here and that terminology is this you see something like 2x here and you'll see it referred to often in class The coefficient is the number out in front of our variable. The variable is the item that can change. It can vary.
The coefficient is just the number that sits out in front or behind that. So the variable is the item that can be replaced. And the last problem, the variable was x.
It's called a variable because it could vary problem to problem. Sometimes we might put a 7 in there. Sometimes we might put a 12 in there. We might substitute an 8. or a 5.5 or a pi squared, something of that nature. But it can vary.
The coefficient is a constant number. Now, 2x and 3x are called like terms. Like terms.
They are called like terms because they have the same variable and those variables have the same exponent. You will hear the words like terms many, many times in this class, so the sooner we get used to dealing with them, the better off it's going to be for us. Let's keep on truckin'.
Alright, gang, something you might not like. Can I really do this? Yes, I can. We're gonna go with word problems, so get yourself super excited. Alright, let's take a peek at what one of these looks like.
So, let's take a peek here. Let's read this problem for a second. Alright, so, you want to buy either scented lotion or bath soap for eight people.
The lotions are six each and the soaps are five each. Write an expression for the total amount you must spend. So, here's the deal. You've got only enough information here.
They don't tell you... the total yet they just tell you what a buy soap for eight people lotions are six dollars and bath soaps are five dollars so in this instance right now I've only got information to write one equation so what I want to do I don't know I like soap so let's use a variable can I pick any variable I want sure I'm gonna label S for soaps now I wanna find a way to relate soap and lotion Now, this is a little bit tricky. The thing is, how many people am I buying items for? Eight people here.
So, how many get soaps and how many get lotions? Well, if there are eight people, how many get soap? I don't know.
S. How many people get lotions? Well, it'd be eight minus the number of people that got soaps, right?
So, if five people get lotions, three people get soap, because that adds to eight. If two people get soaps, then six people get lotions. Now, I need to incorporate that somehow into an equation using money. Well, how much are lotions?
Six dollars. How much are soaps? Five dollars.
So, how many people buy soap? I don't know. S times five dollars would give me my soap cost.
Now, why do I get this here as an equation? Well, think about this. Five times the number of soaps, because I don't let people buy soap, but it's $5 per person. Now, lotions are $6, and that is eight, which is the total number of people, minus soaps. Now, if you wanted to, you could simplify this down as such.
Now, what would I do with that? Now, let us say that five people get lotion. how much does that cost you if five people get lotion how many people have soaps 3 thank you imaginary voice if you think about it if five people get lotions consider 8 total that leaves 3 people get soap and I simply just put in 3 for my s here and it cost forty-five dollars now thus ends what we do you can go henceforth and work on the lesson comment learning