Algebra

Multiplying monomials | Algebra I | Khan Academy

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In this video we will multiply monomials. Let me give you an example of a monomial. For example, 4x 2 is a monomial or a monomial. Why? Mono means one and refers to the number of members. So 4x ^ 2 the whole is one term. In this video we will deal with things like this. What are we not going to deal with? Well, for example 4x ^ 2 + 5x. How many members are there? 4x ^ 2 is the first term, 5x is the second term, so this is not a monomial. This is called a binomial or binomial because “bi” means two. Like your bike has two wheels for example. You can skip to the next videos, if you are ready for two members. We will now multiply the monomials one by one. So we can choose an example to look at. At the end of this video it will be very easy for you to multiply a monomial 5x ^ 2 on this one-term. And I’m actually just going to give you the answer here. And then I’ll slowly take you through a few other tasks, which will lead us to this. But the answer to this is 20x ^ 8. 20x to the eighth degree. Take a look at it and see if you can see a pattern. What do we do with 5 and 4 to get 20? What do we do with 2 and 6 to get 8? This is a little ahead of our reasoning. Before we dive into arithmetic, let’s recall some of the properties of power indicators. A very specific property of degrees you should already be familiar with it. If we look at 5 squared by 5 to the fourth power, what will it be equal to? What if you remember the property of degrees, here we will have a quick negotiation, we always collect the exponents. So 5 squared and 5 squared is equal to 5 to the sixth power. How about 3 to the fourth power 3 to the fifth degree? Again, we always collect the exponents. 4 plus 5 is 9, so we have the 9th power, and base 3 remains the same. It’s great if you remember that, now we are really ready to start to multiply monomials that are something new to you. And the new thing here is that it will we have variables involved. So, let’s get started. Let’s look at the two monomials here. The first monomial is 4x, and the second is only x. The four – I have no other number to multiply, so i only get 4. Can I simplify x by x? Well this is equal to x ^ 2. Remember that if I only have a variable and I have no exponent, this is equivalent to having one, x ^ 1, so x in the first degree in x in the first degree, I add the degrees, as we talked about a while ago, and one plus one is equal to two. Great, let’s move on to another task. If I have 4t by 3t. Okay, four by three is going to be equal to 12, so I combine the odds. And then t by t, again, we imagine that there is a degree one, and will be t squared. So the answer here is 12t squared. So, let’s move on. Once you understand their rhythm, they will become completely clear to you. What will we say if we have 4p on the fifth degree in … say 5p to the third power. What will this be equal to? You will notice a model here, which I always use. It is that I always multiply the odds first. So 4 over 5 is going to be equal to 20. And I always collect the exponents. So p on the fifth and p on the third is p of the eighth degree. Therefore I multiply 4 by 5 and get 20; and I add 5 and 3 to get eight. And if you really want to know why, let’s delve into that and to decompose this first article 4p ^ 5. I can write it as 4 by p, by p, by p, by p, by p, these are five of them. These are the number four and five p’s. And then I can write the second article as by the number five by p, by p, by p. I will now group the numbers, because I can work with them, so let’s put 4 by 5 at the very beginning. And then it’s just a matter of how many p’s I have? We will put them all together too. I have 5 p – these are the first five, and then I have three more. And we can simplify this amazing looking expression, by simply multiplying 4 and 5 to make 20. And then we will present this with a degree. This is the good thing about degrees, that’s why we have them. We can write an unusual expression like this as p on the eighth. And you’ll notice that this is, which we received the first time. So, good. How about 5u on the sixth minus 3y to the eighth degree? Again – we multiply the coefficients and add the power indicators, and I get a simplistic expression. Let’s really get excited, have some fun. We noticed the pattern, so let’s have some fun. I’m just saying I can do more. Minus 9x in the fifth degree minus 3 … Here we use brackets. When there is a minus sign in front, always use brackets. Let’s say x to the 107th power. If I had shown you this before, you would say, “For God\’s sake, there’s nothing I can do, I’m trapped, there is no way out. ” But now you know it’s very simple if you follow the rules. I will multiply the odds, –9 to –3 is 27. Two negatives make a positive, and 9 out of 3 is 27. I will add the degrees. 5 plus 107 is a hundred … oh, not two, I almost made a mistake. Let’s get rid of this, give me a second chance here. Life always gives a second chance, 5 plus 107 is 112. And thus this unusual expression, which is two members – this is the first, this is the second when we multiply and simplify, we get another monomial, which is 27x to the 112th power. I will keep you in suspense until the end. That’s why I’m going to show you a task. What variable to use? You’ve noticed that I’m trying to change variables, to show you that it doesn’t matter. This is an ugly 5, let me get rid of it. Give me another chance with him too. Let’s look at 5x to the third power, 4x to the sixth degree. And I will show you the wrong answer. I had a student who I asked to solve this and this is the wrong answer he gave me. He told me 9x on the 18th degree. This is completely wrong. What did he do wrong? I want you to think for yourself, what were we talking about What did he do with 5 and 4 to get 9? What was he supposed to do? What did he do with 3 and 6 to get 18, and what was he supposed to do? This is the multiplication of monomials by monomials.

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