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

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

Thanks.

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2 Answers

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

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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.

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