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Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$

I need help solving this question. Every reply is appreciated.

Thanks!

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2 Answers 2

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We have:

$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$


Then we will square the fraction: $$ = \sqrt{\frac{(\sqrt{45}+\sqrt{18})^2} {(\sqrt{7+2\sqrt{10}})^2}}$$


Finish the squares: $$=\sqrt{\frac{45+18\sqrt{10} + 18} {7+2\sqrt{10}}}$$


Simplify: $$=\sqrt{\frac{63+18\sqrt{10}} {7+2\sqrt{10}}}$$


Factor: $$\sqrt{\frac{9(7+2\sqrt{10})} {7+2\sqrt{10}}}$$


Crossing Out: $$\sqrt{9}$$


Which is equal to:

$$\sqrt{9} = 3$$

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Hint:

$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}} = \frac{3 \sqrt{5}+3\sqrt{2}}{\sqrt{\left(\sqrt{2}\right)^2+\left(\sqrt{5}\right)^2+2\,\sqrt{2}\sqrt{5}}}$$

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    $\begingroup$ Wish the downvoter had left a comment why. $\endgroup$
    – dxiv
    Feb 4, 2017 at 4:31
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    $\begingroup$ Nice Hint. (+1). And who are these nefarious down voters? $\endgroup$
    – Mark Viola
    Feb 4, 2017 at 5:36
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    $\begingroup$ @dxiv \ Don't worry about it, man. There are people in this world who pride themselves on being walking, talking toothaches. $\endgroup$ Feb 4, 2017 at 7:29

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