in-13-24-divide-and-express-each-quotient-in-simplest-form-in-each-case-list-any-values-of-the-var-2

Question

In $13-24,$ divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{12}{a} \div \frac{6}{4 a}
$$

Alisa Lu · Accepted Answer

So for this question, were given a fraction 12 over A and were asked to divided by six over for a. So before I begin, I&#x27;m going to say that in this problem cannot be zero, as that would make the denominator equal to zero. And you can&#x27;t divide by zero. Okay, So divided by a fraction is the equivalent operation as multiplying by the reciprocal of the fraction that you&#x27;re dividing by. So this problem told they divided by 648 be rewritten as 12 over a times for a over six. And doing this well, before I multiply across, I&#x27;m going to do some simplification, but and cancellation. So this six will cancel to one, and this will become a two. And here the A&#x27;s will cancel out, and we&#x27;ll just be left with four. Okay, given this will have two times for which is eight over one times one, which is one and eight over one, is just eight. So we have a final answer. It

Alisa Lu · Answer

Okay, So for this question, were given a fraction. A square over a divided by three A over four, and were asked toe do this operation will Also specifying values of a which just couldn&#x27;t work. And those values would be a cannot be zero, as that would make the denominator equal to zero. And we can&#x27;t divide by Sarah. Okay, so before it melts Well, first, um, divided. My fraction is the exact same operation that&#x27;s multiplying by the reciprocal of the fraction you want to divide by. So this could be rewritten as a squared over eight times the reciprocal of the V A over four, just four over three A. And so now go ahead and multiply it cross. Before doing that, I&#x27;m going to do some cancellation simplification. So I notice here that this four and this eight will cancel to give one and two. I also notice that this a in this topping hopes to cancel Eso es will cancel out one of these days and be left with one a. Now, I noticed that this a in this a will also cancel. So this entire ages becomes a one and then way have this um, canceling with to say so. This will also become a one. And so in the end will be left with one times one my collar over two times three, which is 16 where a cannot be.

Derrick Hanson · Answer

so for me personally, any time I see that I have a rational expression as part of my problem, any time that I see have a fraction is part of my problem. I prefer to look at everything on the same level. So I like to have everything representatives fraction because something like this b plus three expression that can cause problems. If you&#x27;re not careful, it&#x27;s purposely not being shown us a fraction just so they can see if they can catch you. Remember if some everything can be expressed as a fraction. So if it doesn&#x27;t look like it&#x27;s a fraction, you can just put it over one, all you have to do is put it over one for it to be a fraction for to be irrational expression. Okay, so if we do that now, I think it becomes much clear how to handle our next step, which is to change this from division to multiplication. Remember, the way you do that is you keep your first fraction exactly the same. Change your division to multiplication, and then you flip or find the reciprocal of your second fraction. Now that I&#x27;ve written it over one, it&#x27;s very clear. I think at least that the reciprocal would be won over B plus three. If you don&#x27;t put that one there, though, it can cause you some issues, I think. OK, so I would just recommend doing that to begin with. Either way, once you&#x27;ve taken care of fighting the reciprocal of your second fraction that changes to multiplication. Now it&#x27;s time to look at each of these individual numerator is and denominators and see if we can factor them down before we start simplifying. Reducing looks like, really the only one that&#x27;s going to be able to simplify is four b plus 12 Can factor, I mean, is gonna be four b plus 12. Four B and 12. Both have a four in common, so let&#x27;s pull that four out. Four. Be divided by four would be be plus 12 divided by four. Would be three Be for my denominator is just that a B one for my numerator is just that a one and B plus three is just that as well. So if I&#x27;m looking for things that are the same between my numerator and denominator, there&#x27;s not a whole lot here. I have the quantity of B plus three in my numerator. And I have the quantity of B plus three in my denominator, so those can cancel out. Other than that, I don&#x27;t see anything else that matches up. So I&#x27;ve got four on top my numerator, because four times one is four, right? And then I just have the be on the bottom. So four Overbey Looks like that&#x27;s our final answer. We do need to find any values that could cause this to be undefined. So remember, you get undefined when zero is in the denominator on the bottom. So I&#x27;m looking at my denominators. That would be be right here, and B plus three right here. Okay, These two fractions here, these two fractions here are just repeated. So I have to solve it out twice. Right? So for my first denominator, B is not allowed to be 00 well, you know, that&#x27;s already solved that awesome. So B does not equal zero, but also be plus three is not allowed to equal zero. To solve that out, I would need to subtract three to the other side and I find that be while not being allowed to be zero is also not allowed to be negative. Three. So my answer is four. Overbey as long as B does not equal zero or negative three.

Derrick Hanson · Answer

this is a division problem. So first thing we want to do here is change it into a multiplication problem. Remember, the way we do that first fraction stays exactly the same. We don&#x27;t do anything to change that. We keep that exactly the same. Then we&#x27;re going to change this to multiplication. And the way we&#x27;re going to do that is by flipping or finding the reciprocal of the second fraction. Meaning we will change it to 10 c over three. Be reciprocal. Meaning we just flip it. Okay, now that we have changed it from division to multiplication now we can actually go about trying to factor each of my enumerators and denominators and then cross out anything in common. So six can be factored 22 times three and the bees just there. Five cannot be broken down at all. So we have a five Tennessee 10 can be factor down to two times five. We&#x27;ve got to see three cannot be broken down at all beat. Now that I fully factored each part, I&#x27;m gonna start looking to cancel things out. Anything in the numerator that matches anything in the denominator can cancel out me in between top and bottom. For example. I see that I&#x27;ve got it a three in my numerator and I have a three in my denominator so those can cancel. I noticed that I have a five in my numerator and a five in my denominator so those can cancel. I also see that I have a bee in my numerator and a B in my denominator those can cancel. And finally I have a C and my numerator and to see in my denominator remember, you can Onley cancel between numerator and denominator top and bottom meaning? Yes, I do have two twos, but they are both in their numerator, so they&#x27;re both left behind two times two is four. It looks like I cancelled out all of my denominators, so that would leave me with just a one four divided by one means that my true answer is really just four. Now we&#x27;re not done here because I do have to find any values for variables that would cause me to make this fraction undefined. With division problems, there is a little bit of a catch here. We have two different parts we have to look at. We have to look back at the original equation right here. But we also have to look right here because while we didn&#x27;t change the first fraction, remember, we did find the reciprocal. We did flip that 2nd 1 So now I have to consider this See, as a possible problem, we have to consider this see as a possible problem. And when we flipped the second fraction found its reciprocal. Now we have to worry about this. Be so The question is, let&#x27;s look at five. See first, if we have five c, we cannot have that equal zero. So what value of zero would be a problem? Well, divide five to the other side, and we see, pretty simply see, is not allowed to be zero right 10 c. My other denominator would give me the same problem. Divide by 10 and we just find that C is not allowed to be zero. So we&#x27;ve established that four is our answer, but also see cannot be zero or else it causes this whole thing to be undefined. We also have this three B though that is not able. That does not equal zero. Well, if you divide three over, then you get be does not equal zero, so B and C cannot equal zero. But if they don&#x27;t, our answer is four.

Alisa Lu · Answer

So for this question, were asked to provide a squared minus one by a fraction a over to a plus to a plus two over a. But dividing my fraction is the same. It&#x27;s multiplying by. It&#x27;s reciprocal. So this gives us a squared minus one times a over to a plus two So you can go ahead and say that this is a squared my SWAT over. So let&#x27;s go ahead and factor the numerator here using difference of two squares that gives us a plus one kinds a minus one all over, one times a over background to to a plus one. So I&#x27;m going to go ahead and say that our A in this problem cannot be equal to negative one, as that would make that would make the denominator equal to zero s. So let&#x27;s go ahead and, um, multiply or before I multiply cross, I&#x27;m going to do some cancellation. S 00 Also a cannot be equal to zero because it would make the denominator equal to zero of the original fraction. Okay, so I&#x27;m going to do some cancellation here. The&#x27;s A plus ones cancel out. And so now we&#x27;re just left with a times minus one over one times two. So eight times a minus one over one times 21 times two is just equal to two. So we can go ahead and say that the final answers a times a my swan.

Alisa Lu · Answer

Okay, So for this question, were given fraction a square plus eight a plus 15 over or a and were asked to divided by from one over a plus three divided by history. So dividing by a plus three is the same as multiplying by the reciprocal day plus three. Just one over a plus three. I&#x27;m so come here. I&#x27;m going to say that our A cannot be equal to zero or negative three, as these values would make the denominator equal to zero when you can&#x27;t divide by zero. So let&#x27;s factor out the numerator here into a plus three and a plus five over for a times one over a plus three. So the eighth Mysteries will cancel. And that just leaves us with one times a post five, which gives you a plus five over for eight times one, which is just for a and so that&#x27;s your final answer

Problem

In $13-24,$ divide and express each quotient in s…

Problem 14 Easy Difficulty

In $13-24,$ divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{12}{a} \div \frac{6}{4 a}
$$

Answer

$8, a \neq 0$

Related Courses

Algebra 2 and Trigonometry

Related Topics

Discussion

Top Algebra Educators

Anna Marie V.

Caleb E.

Maria G.

Michael J.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

Ann Xavier Gantert

Algebra 2 and Trigonometry

Anna Marie V.

Caleb E.

Maria G.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$8, a \neq 0$

Algebra 2 and Trigonometry

Anna Marie V.

Caleb E.

Maria G.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Anna Marie V.

Caleb E.

Maria G.

Michael J.

Ann Xavier Gantert

Algebra 2 and Trigonometry