rationalize-the-denominator-and-write-each-fraction-in-simplest-form-all-variables-represent-posi-18

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{4}{4+\sqrt{7}}$

M S · Accepted Answer

So we have four over four, plus the square root of seven. So because we have this radical on our denominator, we need to get rid of that. And the only way to do that is so multiply by its conjugated. So congregate is whenever you have two terms in this case, will have a plus B if the most probably the contrary it we need to multiply about those same two terms would change the sign. So whenever way boiled is, we&#x27;ll get the first term square, minus the seconds for him squared. You could boil it completely, but just note at the middle terms will kill so well because you have a plus and minus. So you just end up with a square minus B squared. So it&#x27;s just easier if you memorize this formula here. So let it go ahead and well supplied by its conjugated. So we have four minus is great of seven over four, minus the square to someone it&#x27;s on the top will have fuller from Spore minus 37 over four squared minus route seven. It&#x27;s weird. So the top of going to go ahead and boil that I&#x27;m gonna go ahead and distribute that for four times 4 16 in the now just before seven. On the bottle, four square to 16 and scared of seven square. It will give us seven well 16 minus four, Route seven over nine.

Ashley Volpe · Answer

So we&#x27;re gonna rationalize and simplify. And so the bottom has a binomial. So we&#x27;re gonna multiply then and by the contra git. And my stylist is not working. So there we go. Four plus radical. Seven over four plus radical. Seven. Now, because we have a binomial on top is Well, we&#x27;re gonna do some foiling. So six times four is 24 six times radical. Seven is six radical seven if four times negative. Radical seven. So this is negative for radical seven, and then we have negative terms. Positive. So that&#x27;s negative. And then radical seven times radical seven is just seven in the denominator. We have four times four, which is 16. We have war times radical seven. We have four times negative, radical seven, and then again, negative times. Positive is negative. And we have radical seven times Radical seven, which is seven. Okay, I noticed that. These will cancel out. I&#x27;ve got some light terms here and here. So when I write my answer, I want to make sure I combine those like terms. So my numerator 24 minus seven, it&#x27;s gonna be 17 and then I have six radical seven minus four radical seven. So that&#x27;s positive to radical. Seven in the denominator. The only terms I have left are 16 and seven, so 16 minus seven is nine, and there is my final answer.

Ashley Volpe · Answer

question five, We&#x27;re going to you take this expression in rationalize and simplify it. So to do this, we want to remove this radical too. That&#x27;s how we rationalize. And we&#x27;re gonna do that by multiplying the numerator and the denominator by radical, too. When we do that, the top is gonna become four radical, too. And the bottom is gonna be radical two times to just for work. Radical four is just too now looking at that, I can notice that four. And that too. So I&#x27;m gonna simplify now to goes into four twice that it goes into 4 to 2 once and that&#x27;s going to give me then too radical, too over one. And since we&#x27;re over one, we don&#x27;t really need that and said Too radical, too becomes are rationalised and simplified. Answer

Ashley Volpe · Answer

and ask us to rationalize and simplify. So to rationalize, we gotta remove this radical three from the denominator. And we can do that by multiplying the top and the bottom by that radical. And so we&#x27;re gonna mult by the numerator and the denominator by radical three. On the top, we get four radical three on the bottom. The two was just gonna come across and then we can multiply the two radicals together. Radical three times Radical. Three is radical. None. Okay, on the bottom, we can simplify. We know that radical non is just three. And so on the top we get four radical three on the bottom. We have two and a three with two. Tom&#x27;s three. So on the bottom, we have six. So now it&#x27;s rationalized, but we need to simplify it. Four and six can be simplified to goes into four twice to goes into 63 times. And so our final answer then will be to radical three over three

Ashley Volpe · Answer

So we&#x27;re gonna rationalize this problem. We have a binomial. So we&#x27;re gonna multiply by the college. You get seven plus radical. You do that on the top and the bottom. Since the top is already about no meal, we&#x27;re gonna have to foil. And so when we do that, we get seven times seven, which is 49. We get seven times radical five. We get another seven times radical, and then we get radical five times Radical five, which is just okay in the denominator we get seven times seven is 49 seven times radical, five, seven times negative, radical five, and then negative terms. Positive is negative and radical. Five times radical. Five is just about So now we need to combine some light terms and clean it up a little bit. So if we combine our like terms 49 plus five, right, so we do 49 5 we get 54 54 and then we have these, like, terms. So if we combine those together, we get seven plus 7 14 radical by and then in the denominator, noticed that we have seven radical five and negative seven Radical five. So those were gonna cross out. And then we have the like terms of 49 negative five. And so if we combined 49 minus five, we get 44. Now I can look at that and tell you that the coefficient of 14 54 and 44 all even numbers so we can simplify this further. Let&#x27;s divide all of those numbers back to we do that. We get 27 we get seven and we get 22. So our answer, then once we do that simplification on the top of this problem, we have 27 plus seven, radical five. And in the denominator of this problem, we have 22. There is the answer to our problems.

Ashley Volpe · Answer

So they want us to rationalize and simplify this. Since in the denominator, I have a binomial, I&#x27;m gonna multiply then by the conjugated. So that&#x27;s gonna be radical. Seven monos too. And we&#x27;re gonna do that in the numerator and in the denominator. The top of this problem, then, is gonna become a distributive property, non radical seven and then non times too. So minus 18 in the denominator, we have a binomial times of binomial. So we&#x27;re gonna foil. So we get radical 49 we get negative too radical. Seven, We get positive to radical seven, and then we get negative two times positive two, Which is negative for work, negative to radical seven and positive to radical. Seven are gonna cancel radical 49 7 So this becomes seven minus four in the denominator. And then we&#x27;re gonna move to our answer and what&#x27;s clean that up a little bit? We&#x27;re gonna end up with non radical seven minus 18 over seven months, for which is three noticing that my coefficient of nine, my 18 in my three are all divisible by three. I&#x27;m gonna simple by that out says I&#x27;m gonna end up with the denominator of one. I&#x27;m not gonna have a denominator when I finish this question. And so my final answer then will be three radical seven minus six.

Problem

Rationalize the denominator and write each fracti…

Problem 20 Hard Difficulty

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{4}{4+\sqrt{7}}$

Answer

$\frac{16-4 \sqrt{7}}{9}$

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Algebra 2 and Trigonometry

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$\frac{16-4 \sqrt{7}}{9}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry