rationalize-the-denominator-and-write-each-fraction-in-simplest-form-all-variables-represent-posi-36

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{3}{x+\sqrt{2}}+\frac{5}{\sqrt{x}}$

M S · Accepted Answer

So we have three over X plus square to plus five over squared. It&#x27;s so because we have these radicals in our denominator here in here, we need to get rid of those. So from the first for action, we need to multiply by its conjugated, so to congregate. In this case, we have a plus B, so it&#x27;s congregate would be used in the same two terms. We&#x27;re gonna change the side, so we&#x27;re gonna have subtraction in the middle. This turns into a squared minus be screwed. When we finally working out and simplify, it&#x27;s easier if you had memorized is conjugated formula they don&#x27;t have to boil every single time. So for first fraction, we&#x27;re gonna multiply the talking bottom bar the contra get someone to move this over. This would be X minus route to over X minus root, too. And our second fraction we still need to get rid of that radical. But we don&#x27;t have any additional subtraction sign. So we&#x27;re just gonna multiply the top and bottom by room picks. So the first fraction will have three times x minus root, too, over X squared minus root. Soon squared plus and then our second fraction will have five x over re x times re X, and I&#x27;ll just give us X. It&#x27;s like a simplify first fraction. Further, by distributing that three so three X minus three root, too over an X square just stays the same as good of Sue Square gives us, too. So X squared minus two and our second fraction is also simplified to its further 16.

Ashley Volpe · Answer

they want us to rationalize three over radical five months to this is a binomial. So we&#x27;re gonna multiply by the conjugal it, and we&#x27;re gonna do that in the numerator and the denominator on the top of this problem. Then we get a distributive property. So three times radical five and then three times two is six in the denominator, we&#x27;re gonna have to foil. So we have radical 25 plus two. Radical five. Modest, too radical. And then we have modest two times positive to Omanis for worth right away. I noticed that I have liked terms in that denominator. Positive to negative two. I&#x27;m gonna cancel out. Radical 25 is just five and so on the denominator of this probably have five minus four. What is our answer, then? Gonna look like in the top. We have three radical files, plus six. Since the denominator is gonna be five minus four or one, we really don&#x27;t need that one under there. And so we&#x27;re gonna leave that as our final answer

Ashley Volpe · Answer

they want us to rationalize and simplify So we&#x27;re gonna multiply. We noticed this binomial. So we&#x27;re gonna have to you multiply by the conjugated We&#x27;re gonna do that in the numerator and in the denominator on the top, we&#x27;re gonna end up with the distributive property. Now, hopefully is you&#x27;ve been going through these problems were starting to recognize that if you multiply our radical by itself, you actually end up with what&#x27;s underneath that radical so radical five x times radical five X is actually just five X and then we&#x27;re gonna do radical five x times two In the denominator, this becomes the foil property. So we&#x27;re gonna most by radical five x times radical five x, which will just give us biotics. We&#x27;ve got a positive to radical politics. We&#x27;ve got a negative to radical five X and then we have negative two times two, which is negative for work. Your inner and outer terms here positive to negative two will cancel. And so you&#x27;ve got five X minus four in the denominator. So nothing else I can see is gonna simplify in this case. So let&#x27;s rewrite our answer. So our final answer and this question is going to be five x plus two radical five x over five eggs minus four

Ashley Volpe · Answer

question 19 were asked to rationalize and simplify. And so we have a binomial, which means we need to multiply by the contra kit in order to rationalize. So that is the first thing I&#x27;m going to do. Remember, the conjugal is the same binomial with the opposites on on the top, this becomes distributive. So I&#x27;m gonna distribute next three times. One is going to give me three and three times negative radical. Five is gonna give me negative three radical five in the denominator. I&#x27;m going to have a binomial times of binomial, so I&#x27;m going to foil one times One is one one times negative radical. Five is negative radical. Five, One times positive radical. Five is gonna be positive. Radical five, and then I have negative radical five times positive radical. Five for negative. Radical 25. Now look and see what&#x27;s going on here. I noticed immediately these, like, terms here, So negative radical five and positive radical. Five. Okay, until out, radical 25 is five. And so this bottom is one monos five. So what is my answer? Gonna look like? As I finished this question up My top is three minus three. Radical five. My denominator is negative for and that is my bond laser

Ashley Volpe · Answer

All right, so we got to rationalize this and simplify in the bottom of this problem in the denominator, we have a binomial. And so to do the rationalization part of this, we&#x27;re going to multiply by something we call the con. Trick it. And the congregate looks just like the binomial that&#x27;s already there. Except that this sun in the center is the different. So since this is three plus radical five, we&#x27;re gonna multiple of a three monness radical five. So on the top part, that&#x27;s not a problem. We&#x27;re just gonna go straight across and we&#x27;re gonna get three minus radical above in the denominator. However, we have binomial times about no meals. So we&#x27;re gonna use something that we can call Boil. And so that means that we&#x27;re gonna most by the first terms together, which gives us non we&#x27;re going to multiply the outer terms together, which gives us monos three radical file. We&#x27;re gonna multiply the inner terms together, which gives us positive three radical five, and then we&#x27;re gonna multiply the last terms together, which is gonna give us negative radical 25. So we&#x27;re gonna do that boiling in the denominator now we started looking at what we have and seeing if there&#x27;s anything else we can do to this problem. Negative three. Positive. Three. Those terms Air gun aggress out radical 25. Well, that&#x27;s just five. And so on the bottom. In the denominator of this problem, we have a nun minus five, which we know is four. So to finish this problem, but the top is gonna be three Monness radical five. And the denominator is gonna be not a minus five, just four.

Ashley Volpe · Answer

all right. So they want me to rationalize this and I see about no meal. So I&#x27;m gonna multiply about con jacket and since I&#x27;m gonna multiply the denominator also gonna multiply the numerator. The numerator here is already about no meal, so I&#x27;m gonna have to foil this problem. So first to radical X times radical X is just gonna be two x two radical x times Negative radical y is going to be negative, too radical. And then you can just put the X and the Y under the same radical sign inner side. We&#x27;re gonna have negative five And then again, that radical X y and then your last terms. You have a negative times a negative, which makes positive, and you have a five and then you have radical wide times radical. Why? Which makes just simply a while when you do that last one There in the denominator we have radical X times radical X, which is just X. We have radical X times with radical y. So that&#x27;s negative Radical X y, inside x and y So that&#x27;s positive Radical X y. And at the end we have a positive and a negative. We have radical wide times radical. Why? Which is just why I noticed these air going to cancel. And then I see here that I have some light terms, So I wanna make sure I combine those in my final answer. So in the numerator, we&#x27;re gonna have to x What? That was to us. We have modest two and minus five, so that&#x27;s minus seven Radical X Y, and then plus five wide the end in the denominator. Since those inner and outer terms canceled, then all we&#x27;re left within our denominator is X minus y. There is our final answer to this question.

Problem

a. Write each fraction in simplest radical form. …

Problem 38 Hard Difficulty

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{3}{x+\sqrt{2}}+\frac{5}{\sqrt{x}}$

Answer

$\frac{3(x-\sqrt{2})}{x^{2}-2}+\frac{5 \sqrt{x}}{x}$

Related Courses

Algebra 2 and Trigonometry

Related Topics

Discussion

Top Educators

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

Ann Xavier Gantert

Algebra 2 and Trigonometry

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\frac{3(x-\sqrt{2})}{x^{2}-2}+\frac{5 \sqrt{x}}{x}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry