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Friday, February 2, 2024

2024/009) Find \sqrt{\sqrt{9} - \sqrt{8}}

We have 8 = 2 * 4 = 2 * 2^2

so    \sqrt{8} = 2 \sqrt{2}

now \sqrt{9} = 3

so   \sqrt{\sqrt{9} - \sqrt{8}}

\sqrt{3 - 2\sqrt{2}}

this is of the form  \sqrt{n - 2\sqrt{n-1}} when n = 3

so  \sqrt{3 - 2\sqrt{2}} = \sqrt{2 + 1 - 2 * \sqrt{2} * 1}

= \sqrt{(\sqrt{2})^2 + 1^2 - 2 * \sqrt{2} * 1}

= \sqrt{2} - 1                                                                                                                                                                                                                           

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