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Simplify :

$\sqrt{10+6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}}$

I have tried completing square but failed, Can anyone help me please? Thanks.

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Perhaps you could write $a+b\sqrt2+c\sqrt3+d\sqrt6=\sqrt{10+6\sqrt2+5\sqrt3+4\sqrt6}$, square both sides, set like coefficients equal, and try to solve for $a,b,c,d$. –  Gerry Myerson Jun 15 '13 at 12:34
    
@GerryMyerson Thank you. This is a good idea. –  mathe Jun 15 '13 at 13:26
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2 Answers

up vote 2 down vote accepted

$$10+6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}=5(2+\sqrt3)+2\sqrt6(2+\sqrt3)=(2+\sqrt3)(5+2\sqrt6)$$

Now, $$5+2\sqrt6=3+2+2\cdot\sqrt2\cdot\sqrt3=(\sqrt3+\sqrt2)^2$$

and $$2+\sqrt3=\frac{4+2\sqrt3}2=\frac{3+1+2\cdot\sqrt3\cdot1}2=\frac{(\sqrt3+1)^2}2$$

Can you take it from here?

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Very nice work! Thank you, I see. –  mathe Jun 15 '13 at 13:34
    
@maplematica, my pleasure. –  lab bhattacharjee Jun 15 '13 at 13:34
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Hints:

$$10+6\sqrt2+5\sqrt3+4\sqrt6=2(\sqrt2+\sqrt3)^2+6\sqrt2+5\sqrt3\;\ldots$$

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could you please elaborate a bit? –  lab bhattacharjee Jun 15 '13 at 13:29
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