# How to solve $\sqrt{9-4\sqrt{5}}=$?

Need some hints how to solve this: $\sqrt{9-4\sqrt{5}}=$ ?

Thanks.

• roots is about the zeros of a function (see it's tag wiki). Use arithmetic for the (square) root operation instead (see the same tag wiki). Please read the tag wiki before applying a tag. – AlexR Feb 2 '15 at 15:10
• @AlexR "radical" is an appropriate tag here, too (in lieu of "root") – Namaste Feb 2 '15 at 15:15
• @amWhy Didn't know such an extremely localized tag existed. Thanks! – AlexR Feb 2 '15 at 15:17

$\sqrt{9-4\sqrt{5}}=\sqrt{5+4-2\cdot 2\cdot \sqrt{5}}=\sqrt{(\sqrt{5}-2)^{2}}=|\sqrt{5}-2|=\sqrt{5}-2$

• I think for the sake of completeness one should mention that $\sqrt{(\sqrt5-2)^2}=|\sqrt{5}-2|$ $=\sqrt{5}-2$, so that one may not be tempted to state otherwise when it is hidden, e.g. if one wrote $$\sqrt{9-4\sqrt{5}}=\sqrt{4-2\cdot2\cdot\sqrt{5}+5}=\sqrt{(2-\sqrt{5})^2} \stackrel{\color{#F01C2C}{\rlap{\small\,\,!}{\displaystyle\triangle}}}{=} 2-\sqrt{5}.$$ – Workaholic Feb 2 '15 at 15:15
• Didn't asked to solve this for me, but ok. I always fail on this type of arithmetic (where you need split units). Anyway, thanks. – gintko Feb 2 '15 at 15:17
• i am sorry creatur , but i hope that u wont have problems in these type of questions anymore – avz2611 Feb 2 '15 at 15:18
• @avz2611 Maybe you should add a small comment of how to find the completing square easily to match up for this. Nice answer, tho. (+1) – AlexR Feb 2 '15 at 15:19
• @Workaholic How did you wizard that triangle there? :D EDIT: nvm $\color{#F01C2C}{\rlap{\small\,\,!}{\displaystyle\triangle}}$ \color{#F01C2C}{\rlap{\small\,\,!}{\displaystyle\triangle}} – AlexR Feb 2 '15 at 15:19

This can be computed by a Simple Denesting Rule:

Here $\ 9-4\sqrt 5\$ has norm $= 1.\:$ $\rm\ \color{blue}{subtracting\ out}\,\ \sqrt{norm}\ = 1\,\$ yields $\,\ 8-4\sqrt 5\:$

which has $\, {\rm\ \sqrt{trace}}\, =\, \sqrt{16}\, =\, 4.\ \ \rm \color{brown}{Dividing\ it\ out}\$ of the above yields $\ \ 2-\sqrt 5$

Remark $\$ Many more worked examples are in prior posts on this denesting rule.

• Note: generally, as here, you may need to change the sign if you want the positive branch of sqrt. – Bill Dubuque Feb 2 '15 at 15:38
• Very interesting. I didn't even know about this simple rule! – rubik Feb 2 '15 at 16:26