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I'm having problems finding the conjugate of $\frac{1}{4\sqrt{3}-7}$

The answer I get is as follows: ${-4\sqrt{3}-7}$

However the answer given is ${-4\sqrt{3}+7}$

Here are my workings out... $\frac{1}{4\sqrt{3}-7} * \frac{4\sqrt{3}+7}{4\sqrt{3}+7} = \frac{4\sqrt{3}+7}{-1} = {-4\sqrt{3}-7}$

I'd appreciate any guidance on where I've gone wrong.

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    How did you get $-1$ for the denominator? And what is your definition of "conjugate"? – 2017-02-12
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    In beginning you say the answer you got is $-4(\sqrt{3}-7)$ but when you show your working you get $-4(\sqrt{3}+7)$ – 2017-02-12
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    Aren't you forgetting a square root for that $\;7\;$ ? – 2017-02-12
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    @kingW3 :-| That's what I was thinking... – 2017-02-12
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    Wait:$$(\sqrt3-7)(\sqrt3+7)=3-49=-46\implies \frac1{4(\sqrt3)}=\frac{\sqrt3+7}{-46}\ldots$$ How is there nothing wrong with his work?? Am I missing something here? – 2017-02-12
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    @DonAntonio Whoops, you saw nothing – 2017-02-12
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    I've cleaned this up a bit now as there were some unnecessary brackets in there that weren't in the original question. Sorry about that. – 2017-02-12
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    @Hemmed Yes...brackets/parentheses in mathematics can make a **huge** difference...! – 2017-02-12
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    What a nasty problem. The student gets caught up in all of the detail of rationalizing the denominator and forgets to take the conjugate when he or she is almost finished. – 2017-02-13

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Well, after all the changes:

$$(4\sqrt3-7)(4\sqrt3+7)=48-49=-1$$

and then indeed:

$$\frac1{4\sqrt3-7}=-\left(4\sqrt3+7\right)=-4\sqrt3-7$$

and yes: the conjugate is $\;\frac1{4\sqrt3+7}\;$ , and what you say is the answer given is neither the conjugate nor the product of the original expression by its conjugate.

Added following an idea by projectilemotion: Perhaps the idea is first to rationalize the expression and then to find its conjugate:

$$\text{Rationalizing:}\;\;\frac1{4\sqrt3-7}\cdot\frac{4\sqrt3+7}{4\sqrt3+7}=-4\sqrt3-7$$

and now the rightmost expression's conjugate indeed is: $\;-4\sqrt3\color{Red}+7\;$ .....tadaaah!

It's hard to tell what they meant without knowing a priori their definitions... BTW, in this case and for me, the conjugate could as well be $\;4\sqrt3-7=-7+4\sqrt3\;$ ...Funny.

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    What I don't understand is that the conjugate I get ends with -7 but in the example given in the quiz the conjugate is actually +7. – 2017-02-12
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    @Hemmed The conjugate **is** $\;\frac1{4\sqrt3+7}\;$, but where you wrote "my working" you are *actually* multiplying the original expression by its conjugate. Two different things... – 2017-02-12
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    Yeah, I've got pretty confused the full example is given here: [link](http://www.mathopolis.com/questions/q.php?id=3195) Is there a language thing where I'm missing what the question is asking for? – 2017-02-12
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    @DonAntonio Interestingly, when rationalizing the denominator of the conjugate $\frac{1}{4\sqrt{3}+7}$, the answer $-4\sqrt{3}+7$ is given. Perhaps that is what the question is referring to. – 2017-02-12
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    @projectilemotion That seems to be a reasonable assumption. I couldn't tell without knowing the *proper definition* the use. – 2017-02-12
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    @projectilemotion I think this might be it. Basically the question i'm given is 'What is the conjugate of...' but then the answer is as follows: The conjugate of (4√3 - 7) is (4√3 + 7), but this is not the answer! [image of full text](http://www.mathopolis.com/questions/latex/1/f/b72fad27ccce19b5b2f26e050a794b1.gif) Unfortunately I'm still now sure how to get to -4(sqrt 3)+7 as the conjugate, as I get -4(sqrt 3)-7 as it. – 2017-02-12
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    Yes sorry, there's a follow up bit which is: Now we can find the conjugate by changing the sign in the middle: [image](http://www.mathopolis.com/questions/latex/b/c/58b7d909fd494b513dcc64acdd31e21.gif) but I get the conjugate as (4√3 - 7) – 2017-02-12
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    @Hemmed Read the stuff I just added to my answer. It fits what you say the answer and it is most probably they way they defined in your school the term "conjugate": after first rationalizing and etc. – 2017-02-12
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    @DonAntonio yes, sorry - this is the definition I knew. This is really dumb, but when you rationalize how do you get to −4√3−7 ? – 2017-02-12
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    @DonAntonio That's a good guess. Or maybe the expected answer was $\frac{1}{-4\sqrt{3}-7}\,$. – 2017-02-12
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    @Hemmed I think we already went over this: $\;(4\sqrt3-7)(4\sqrt3+7)=16\cdot3-7\cdot7=48-49=-1\;$ ... – 2017-02-12
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    @DonAntonio I understand that part, but, and this is probably me being stupid, how does $\frac{4\sqrt{3}+7}{-1}$ become $-4\sqrt{3}-7$ – 2017-02-13
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    @Hemmed ?? Well, that is basic mathematics: $$\;-a-b=-(a+b)=\frac{-(a+b)}1=\frac{a+b}{-1}=\ldots etc.$$ – 2017-02-13
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    @DonAntonio I had a brain fail :( – 2017-02-13