Simplify $\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}$

Question

Simplify $\displaystyle{\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}}$.

I get this when I was doing another Q, but I don't know how to further simplify it. Can anyone help me, please?

Let $x$ be your number divided by $9$. Brun's method mentioned in math.stackexchange.com/questions/152797/… finds the relation $x^4 - 50 x^2 + 125 = 0$ numerically, which suggests $x = \sqrt{25 + 10 \sqrt{5}}$. To actually prove this, the answer by Raymond Manzoni is more appropriate.

Raymond Manzoni · Accepted Answer · 2012-07-01 07:59:46Z

up vote 5 down vote accepted

Hint: Let's note $o:=\frac {1+\sqrt{5}}2$, $a:=\sqrt{10-2\sqrt{5}}$ and $b:=\sqrt{5+2\sqrt{5}}$

then $ab=\sqrt{30+10\sqrt{5}}=5+\sqrt{5}$

Compute $(o\cdot a+b)^2$ to conclude.

edited Jul 1 '12 at 7:59

answered Jul 1 '12 at 7:43

Raymond Manzoni
15.3k22043

Get it! Thank you!~ – ᴊ ᴀ s ᴏ ɴ Jul 1 '12 at 13:40

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