rationalize-the-denominator-and-write-each-fraction-in-simplest-form-all-variables-represent-posi-15

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{1}{1+\sqrt{3}}$

William Greiner · Accepted Answer

how to simplify 1/1, plus the skirt of three. You need to rationalize the denominator. So in order to rationalize the denominator, you have to multiply by the conjugated of one plus the sward of three. The conjugal. It is one minus the square root of three. In your numerator, the one has to distribute to both terms in the numerator. So in this case, one times one is one and one times a skirt of three is the squirt of three. So our numerator is one minus the square root of three. In the denominator, you need to use an extended distributive property. And the one in the first expression needs to multiply by both terms. So that gives you one times One is one one times. Um, the squirt of three is squirt of three. Because the squirt of three is being subtracted, it is technically a negative squirt of three. Um and then the squirt of three in the first expression also needs to be distributed toe both terms in the 2nd 1 So the square root of three times one is a positive squirt of three, and the scores of three times the negative squirt of three is going to be a negative square root of nine when you simplify in combined like terms, Um, the squirt of three. The negatives were 23 plus a positive squirt of three. Um, are opposites. So that creates zero. So you have one minus the square root of nine. And this word of nine is 31 minus three is negative too. So you just created a rational number in the denominator. Okay, Another way to write this is to take the negative from the numerator and make both of the terms. Sorry. Take the negative from the denominator. And to create a positive too. And the denominator, both of my terms and my numerator would have to be the opposite. So the opposite of one minus the skirt of three is a negative one. Plus a positive squirt of three, all divided by a positive too

Ashley Volpe · Answer

question three gives us the expression one over radical three and it ask us to simplify this and write and simplest form. If we&#x27;re going to rationalize this, we have to remove this radical three. So to do that, we&#x27;re gonna multiple the numerator and the denominator by radical three. When we do that on the top, we get radical three and on the bottom, we get radical, non and so radical. Non is three and we have radical 3/3 at the end.

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

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

her. It&#x27;s over this question. We&#x27;re gonna rationalize and simplify. I noticed that my denominator has about no meal. And so to rationalize it, I&#x27;m gonna multiply by its kontic it. And so the Kontic it looks exactly the same, except that it&#x27;s gonna be the opposite sign in the middle. So we&#x27;re gonna multiply about three plus radical three for this problem. Now on top, What we get is distributive. So we&#x27;re gonna do four times three with just 12. And there were 84 times radical three to get four. Radical three on the bottom. We have a binomial times about no meals, so we&#x27;re gonna foil three times three is none. Three times radical. Three is three radical, three negative radical. Three times three is negative. Three radical three. And then we have negative radical three times Radical three to give us a negative radical non. So now that we&#x27;ve done that multiplication, let&#x27;s look and see. Is there anything else going on that we can deal with and so I can see that positive three radical three negative radical. Three radical. Three will cancel out. I know radical non is three. And so on my bottom here on my denominator. I have none on this three. So moving forward from that, what do we get? Her top is gonna be 12 plus four radical three. And the denominator nine minus three is gonna be six. So I&#x27;ve done the rationalization, and I just need to make sure I&#x27;m simplified. My brain immediately shows me I&#x27;ve got 12 4 and six those air all divisible by two. So I&#x27;m gonna divide them all by two to simplify this question. So, what is that gonna give me that&#x27;s gonna give me six plus two Radical three over three. That is gonna be my 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

Problem

Rationalize the denominator and write each fracti…

Problem 17 Hard Difficulty

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.
$\frac{1}{1+\sqrt{3}}$

Answer

$\frac{-1+\sqrt{3}}{2}$

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Ann Xavier Gantert

Algebra 2 and Trigonometry

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$\frac{-1+\sqrt{3}}{2}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

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Algebra 2 and Trigonometry