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I need to solve the following problem:

$$\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=\,?$$

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The solution to $\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=x$ is $x=\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}$. –  David H Apr 26 at 17:09
    
Surely you want to know of to simplify this right? –  user88595 Apr 26 at 17:40

3 Answers 3

up vote 2 down vote accepted

$\frac{7+\sqrt{5}}{7-\sqrt{5}}=\frac{7+\sqrt{5}}{7-\sqrt{5}}\cdot\frac{7+\sqrt{5}}{7+\sqrt{5}}=\frac{(7+\sqrt{5})^{2}}{49-5}=\frac{(7+\sqrt{5})^{2}}{44}$

Then you are taking the root of the quotient (note for $a,b >0$ $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$) and you get

$\frac{7+\sqrt{5}}{\sqrt{44}}=\frac{7+\sqrt{5}}{2\sqrt{11}}=\frac{\sqrt{11}(7+\sqrt{5})}{22}$

None of the answers proposed is correct: we can use the squared value we have calculated

$\frac{(7+\sqrt{5})^{2}}{44}=\frac{3}{11}+\frac{\sqrt{35}}{22}$

As you can see it is not rational, so you exclude $1$ and $2$

Then $(6 \pm \sqrt{35})^2= 36+35 \pm 12 \sqrt{35}$ and you can see that both of them are incorrect.

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It's a multiple choice question, the options are a)6+√35 b)6-√35 c)2 d)1 Is it possible to get one of the above answer? –  SIBHI S Apr 26 at 17:21

Squaring the fraction gives $$\frac{7+\sqrt5}{7-\sqrt5}=\frac{1}{44}(7+\sqrt 5)^2$$ so by taking the square root we find $$\frac{7+\sqrt 5}{2\sqrt{11}}$$

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It's a multiple choice question, the options are a)6+√35 b)6-√35 c)2 d)1 Is it possible to get one of the options as answer? –  SIBHI S Apr 26 at 17:21
    
No one of these options is an answer. –  Sami Ben Romdhane Apr 26 at 17:25

\begin{align} \frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}&=\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}\cdot \frac{\sqrt{7+\sqrt{5}}}{\sqrt{7+\sqrt{5}}}\\ &=\frac{(\sqrt{7+\sqrt{5}})^2}{\sqrt{(7-\sqrt{5})(7+\sqrt{5})}}\\ &=\frac{7+\sqrt{5}}{\sqrt{7^2-(\sqrt{5})^2}}\\ &=\frac{7+\sqrt{5}}{\sqrt{49-5}}\\ &=\frac{7+\sqrt{5}}{\sqrt{44}}\\ &=\frac{7+\sqrt{5}}{2\sqrt{11}}\cdot\frac{\sqrt{11}}{\sqrt{11}}\\ &=\frac{7\sqrt{11}+\sqrt{5\cdot11}}{2(\sqrt{11})^2}\\ &=\frac{7\sqrt{11}+\sqrt{55}}{22} \end{align}

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