rationalize-the-denominator-and-write-each-fraction-in-simplest-form-all-variables-represent-posi-28

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{7+\sqrt{5}}{7-\sqrt{5}}$

M S · Accepted Answer

this is number 30. So you have seven plus street five over seven, minus three pots. So because we have this square root and the denominator, we need to get rid of that. And the only way to do that is to multiply. That&#x27;s hopping bottom by the conjugated. So in this case, we have a minus. B, so it&#x27;s contra get would be using the same terms, but we need to train just on in the middle, so I&#x27;m gonna have a plus B, and whenever we foil this out will have a squared minus and then B squared. It&#x27;s easier if you would just memorized the formula so you don&#x27;t have toe Boyle and expand everything. Formula would just make the problem solving go by quicker. So when will it&#x27;s probably talking bottom by seven plus route drive over seven plus Rupa so have seven plus route by time. Seven place Route five and on the bottom will have seven squared minus five squared. So if you boil, this harp will have seven times seven gives us 49 most bloody used to seven route five animals apart. These two will have seven Route five and then most party soon after. Give this spot. You&#x27;re on the bottom seven. Square gives us 49 and scared about some scary father spot. It&#x27;ll be simple by this. Further, I&#x27;m gonna combined this 49 our five. What an hour plus five gives us 54. And because we have the same based here and here, we can combine their coefficients of seven. So seven plus seven gives us 14 the 54 plus 14 roots all over. But you know, minus five gives us 44th so they can go over the greatest common factor between these three numbers. Let&#x27;s start out with two. So 54 divided about two. That would give us 27. Forcing about about two gives a seven and 44 divided by two. Gives us 22 and that&#x27;s this forest.

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

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.

Ashley Volpe · Answer

question for Ask us to rationalize and simplify five over radical 10. To do that, what I&#x27;m gonna need to do is multiply at the top and the bottom by radical 10. On the top of this problem, that&#x27;s going to give me five times radical. 10. Think and on the bottom I&#x27;m going to get the radical of Tin Toms 10 which is radical. 105 radical 10 radical to 100 is simply 10. Oh, and then I have to remember that I can cross our simplify these so far I will go into five once and it&#x27;ll go into 10 twice so I can actually reduce that fraction down to get a final answer then of radical 10 over t.

Ashley Volpe · Answer

All right, So we want to rationalize and simplify this. This is going to be a binomial. So we need to multiply by the Contra kit. And we&#x27;re gonna do that in the numerator and the denominator on the top, we get a distributive problem. So we&#x27;re gonna distribute four times for to give us 16. We&#x27;re gonna do four times negative seven to give us negative for radical sent me? Yeah, in the denominator, we have about no meal times about no meals. So we&#x27;re gonna foil four times four to get 16 four times negative radical seven to get negative for radical seven, four times positive radical seven to get positive for radical seven. And then we have positive and negative here. So negative radical 49. I noticed that I have liked terms that are gonna cancel out. And I know radical 49 is seven. So my denominator becomes 16. Modest seven. So what does this make my problem look like? So in the numerator, then I have 16 minus four, radical seven, and in the denominator, I have not. And that&#x27;s a far as I can go. Final answer

Ashley Volpe · Answer

rationalize and simplify this. And I like to simplify first and recognize five and 15 as reducing to one and three. So we currently then have the radical 500. And in the numerator and in the denominator, we have three radical, too. To rationalize this, we&#x27;re gonna multiply by that radical that&#x27;s in the denominator. We do that on the numerator and the denominator. So that&#x27;s going to give me on top radical five times radical to which is radical 10. And on the bottom, I&#x27;m gonna get three radical four. Well, radical four is just too. And so this becomes three times to on the denominator side. And so when I do that, my final answer then will be radical 10 over six.

Problem

Rationalize the denominator and write each fracti…

Problem 30 Hard Difficulty

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{7+\sqrt{5}}{7-\sqrt{5}}$

Answer

$\frac{27+7 \sqrt{5}}{22}$

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$\frac{27+7 \sqrt{5}}{22}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry