Need some hints how to solve this: $\sqrt{9-4\sqrt{5}}=$ ?
Thanks.
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[tag:roots] is about the zeros of a function (see it's tag wiki). Use [tag:arithmetic] for the (square) root operation instead (see the same tag wiki). Please read the tag wiki before applying a tag. – AlexR Feb 02 '15 at 15:10
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@AlexR "radical" is an appropriate tag here, too (in lieu of "root") – amWhy Feb 02 '15 at 15:15
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@amWhy Didn't know such an extremely localized tag existed. Thanks! – AlexR Feb 02 '15 at 15:17
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See also: Denesting radicals at Wikipedia, Strategies to denest nested radicals (and the posts linked there). – Martin Sleziak Oct 23 '16 at 23:02
2 Answers
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$\sqrt{9-4\sqrt{5}}=\sqrt{5+4-2\cdot 2\cdot \sqrt{5}}=\sqrt{(\sqrt{5}-2)^{2}}=|\sqrt{5}-2|=\sqrt{5}-2$
avz2611
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1I 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 02 '15 at 15:15
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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 02 '15 at 15:17
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1i am sorry creatur , but i hope that u wont have problems in these type of questions anymore – avz2611 Feb 02 '15 at 15:18
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@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 02 '15 at 15:19
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2@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 02 '15 at 15:19 -
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@AlexR Just a combination of
\triangleand!:\stackrel{\color{#F01C2C}{\rlap{\small\,\,!}{\displaystyle\triangle}}}{=};-) – Workaholic Feb 02 '15 at 15:21 -
@Workaholic Yeah, figured out I might as well see myself in the MathJax context menu. I almost forgot about that since answers can simply be "edited" to check.$$$$The following looks uglier but is less TeX:
\color{#F01C2C}{\displaystyle\triangle\!\!\!\small!}: $\stackrel{\color{#F01C2C}{\displaystyle\triangle!!!\small!}}=$ – AlexR Feb 02 '15 at 15:21
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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.
Bill Dubuque
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Note: generally, as here, you may need to change the sign if you want the positive branch of sqrt. – Bill Dubuque Feb 02 '15 at 15:38
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