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Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator.$ ...

Question

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator.$ rac{3}{t}, rac{8}{t^{3}}$

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator. $\frac{3}{t}, \frac{8}{t^{3}}$



Answers

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator. $\frac{3}{t}, \frac{8}{t^{3}}$

Hey, guys. So let's try to simplify this ratio. Where? Hirsche our way. So for the numerator, what do you guys think would be the common factor here? How about ***? Pretty? All right, so let's fucker it out and then we'll have one minus two t plus t square, right? All right, let's help. Everyone was the denominator. Last factor out seven t squirt. Right, Because all the elements have t scorer and they also have seven to so seventies. Gratified by seventies where it just one negative 14 T cube divided by seven. T square would be negative two t here and plus 72 power Ford about it by seventies where we just give us a T Square. All right, so both the top and the denominator half this tray so less factory. So last cancel it out. And they also have won t to. Okay, so just like this, then t can again cancel each other out in a final answer. Should be negative. Three divided by 70. So, just like that. Great job, everyone. Thank you so much for listening

Running the LCD off to t squared plus 70 minus 15 and t squared t squared plus three t minus 10. So when a factor. Everything here. So can I factor this? 1st 1 to t squared plus 70 minus 15. Let's see, by trying to factor this well to t square, let me to tea times t OK, there, 15. That's five times three. And, um Well, let's see after somehow add to a seven in the middle. So, um, why do five times three Who? But I have a two t here. So if I do who The difference of 10 and three is seven. Put the five here that I get two times five. I get a 10 t for the three here. Any different signs at the edge of positive seven. So I do A plus here at a minus Here are gonna work. Let's check this. Forget to tee times to use to t square like a two teeth. In fact, that plus 10 t minus three t right plus 10 T minus treaty. It's plus 70 and then I get minus three times five of minus 15. So yes, this factors as to t minus three times C plus five. And then this is gonna factor. Let's see. I have Well, I have t square. That's tee times T No, I have 10 while 10. You know his work? 10 times one, But we're five times to who and I have different times two is three. I do five and two at the ad to a positive three's. A plus will be on the five and the minus. Here. That's a T Square minus two T plus five p plus treaty in the minus 10. Okay, so now that everything's factored my LCD, I look at my factors, so let's see, I have Well, I have one. I have a factor of two. T minus tree. I don't have a to t minus three here, so I'm gonna need a factor of two T minus three in my LCD, then let's see. Well, I have I have a t minus two factor in this denominator. I have no t minus two here, so I'm gonna need a factor of T minus two. Now. My last factor is what t plus five and t plus five occurs in both denominators. So therefore I only need one factor of T plus five. So I have a factor of t plus five, and therefore, my LCD is, uh to t minus three times the quantity t minus two times a quantity T plus five. I take your

So we have to rational expressions. Eight divided by nine times P to the dirt and five divided by 12 p to the eight. So once we have this, we're trying to find the least common denominator LCD and to approach this week, start making the factories ation trees for these two denominators. So for 12 p to the gates, we get two times six pleats the eight and two times three p to the AIDS, which is equal to P. Times pizza. Seven. He timed pizza. The six p Times Peed Toothy fifth. We know P occurs eight times just writing it out. To be sure. P to the fourth p Times Pizza that third p Times P to Second and Pete Times Pete. So we know that P occurs eight times eight total times in this accusation tree and we can count for two. It count. It occurs twice. We also shouldn't forget that there is a three. There's a three here, so three occurs one time over here, and then we can move on to the other factor, which is nine p nine p cute. And we know this is equal to three times three p to the there is equal to three times P to the third, and this is equal to p times p squared P Times piece owned. Here we could we know that p his present three times and three is present twice. So with these numbers now, we can create the least common denominator. So the number the all the unique factors R p two and three and the greatest amount of times that to occurs between these two ah factor ization trees is too. So he races his second power. The greatest number time that p occurs is Gates raised its the eighth power in the greatest honor times three occurs is in nine. Pete's cute, which is two times, and we multiply these together we get four times 94 times nine is 36 times p to the eighth. So now we have our answer to the question for the LCD

And he a you by 27 over nine minus Or he's where So my numerator is the difference of cubes. I have to t minus the ready. Same sign, different sign. Always positive. So to tease where is border? He's weird. Three times two is e three squared is nine And in my denominator I'm gonna factor out in negative so that I'm gonna have or he squared minus nine. And when I factor 40 squared minus nine gonna get to t plus three to t minus three. So, like to t minus three, cancel out and I'm left with a negative outfront or e squared Class six e plus nine divided by to be plus three.


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