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In $3-12,$ multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined.$$\frac{2 a+4}{6 a} \cdot \frac{3 a^{2}}{a^{2}+2 a}$$

$1, a \neq 0,-2$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 3

Multiplying and Dividing Rational Expressions

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Missouri State University

Campbell University

Baylor University

Lectures

01:32

In mathematics, the absolu…

01:11

01:41

In $3-12,$ multiply and ex…

01:30

00:42

05:28

02:07

00:36

01:59

01:43

In $13-24,$ divide and exp…

01:09

06:54

In $3-20$ , perform the in…

first things first year. I want to remind myself that to a plus four is a quantity and a squared plus two. A is also a quantity so that I don't try and split those things up. Because if it's addition or subtraction, that is binding those together. Okay, now, after that, I am looking to simplify each of my four different pieces, my two numerator and my two denominators. I want to simplify each them separately as far as I possibly can. One factor each of them separately as far as a possibly can. And then I looked to cancel things out. For example, starting up in the top left my left numerator to a plus four. I can factor that because to a and four both share a common factor, namely to so I can pull that two out to a divided by two would leave me with a plus four divided by two would leave me with two. In my denominator, six can be split up into two and three and then we've also got the A in my numerator. My second numerator, my right numerator three cannot be split into anything, but I can split a squared into a times a right. That's literally what a squared means is a times a finally, my last denominator, my right denominator a squared plus two A. If there is a square term so that means there's a whole bunch of different types of factoring that I could do In this case, though, you should mean well, in every case, you should always check for a G C F first before doing any other type of factoring. And in this case, G cf is actually all it is a squared into a both heaven a in common. So I'm simply going to pull that a out a squared divided by a would leave me with just in a to a divided by a will leave me with two Now that I have fully factored all of my different parts I'm ready to start canceling things. I see that I've got a two in my numerator and a two in my denominator. So those king cancel I see that I have a three in my numerator and a three in my denominator so those can cancel. I see that I have the quantity of a plus two in my numerator and I have the quantity of a plus two in my denominator. So those two cancelled. I see that I have in a and another A and my numerator and I have a and another A. In my denominator, I have two A's, both on top and on bottom. So those canceled. Now this is very interesting because it looks like I have crossed out everything I have here. That's not a problem, though. Just remember when you're multiplying or dividing, which that's what we're doing here. When things cancel in multiplication division, that does not mean zero right in addition or subtraction, cancelling out a zero to minus two a zero. But if you're multiplying or dividing and things cancel, they cancel toe one right two divided by two is 15 divided by five. Anything divided by itself is one. So when everything cancelled here, that's great. That means our answer forgot to change color. That means our answer is legitimately just a nice, simple one. We're not done yet because we do also have to take care of any values, Um, for variables for a that could possibly make this undefined meaning, remember, causes to have zero in the denominator. Well looking, my first denominator, I just have six A. Well, that's pretty straightforward. If a happens to be zero, that would be a problem, because six times zero is zero and the whole thing is messed up because it's undefined. Not a lot of math that you have to do for that. Although I will make this look more like an A over here just to be clear for my second denominator. Now, this one is not quite so clear. Okay, so I'm going to first stuff. Look at my factored version because that would be was best if I have a squared plus to a set not equal to zero. We need to factor that first, right? So again, I can pull in a out, just like I already did earlier in the problem. That's what I'd actually want to be dealing with. Remember, we have the zero product property that says, if you have two things to quiet is being multiplied to each other, but they both equals zero, or the whole thing is post equal zero. Then that means you can set each piece equal to zero separately or, in our case, not equal to zero. So this one's not new. We already knew that a could not equal zero. That's a repeat. So not a big deal. Looking at my second piece, though, I would need to subtract two over, and I would get that A does not equal negative, too. And so that needs to be added over here because that is the two. These are the two values we cannot have or also we, because this fraction to be undefined.

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