Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
What was the other Q? – anon Jul 1 '12 at 7:27
the square of this is solution of a quadratic equation. – Raymond Manzoni Jul 1 '12 at 7:30
Let $x$ be your number divided by $9$. Brun's method mentioned in… 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. – WimC Jul 1 '12 at 10:02
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.

share|cite|improve this answer
Get it! Thank you!~ – ᴊ ᴀ s ᴏ ɴ Jul 1 '12 at 13:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.