\sqrt{31+ \sqrt{936} } - \sqrt{21- \sqrt{416} } =.... Question from @ichaawia

October 2018 2 98 Report

\sqrt{31+ \sqrt{936} } - \sqrt{21- \sqrt{416} } =

NamakuDuck $\sqrt{31+\sqrt{936}}-\sqrt{21-\sqrt{416}}$
=> $\sqrt{31+\sqrt{4(234)}}-\sqrt{21-\sqrt{4(104)}|}$
=> $\sqrt{31+2\sqrt{234}}-\sqrt{21-2\sqrt{104}}$
=> $(\sqrt{18}+\sqrt{13})-(\sqrt{13}-\sqrt{8})$
=> $\sqrt{18}+\sqrt{8}=3\sqrt{2}+2\sqrt{2}=5\sqrt{2}$

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ImmanuelSinaga $$\begin{align} &=\sqrt{31+\sqrt{936}}~-~\sqrt{21-\sqrt{416}} \\ &=\sqrt{31+\sqrt{2^2~\times~234}}~-~\sqrt{21-\sqrt{2^2~\times~104}} \\ &=\sqrt{31+2\sqrt{234}}~-~\sqrt{21-2\sqrt{104}} \\ &=~\text{Ingat~:} \boxed{\sqrt{(a+b)~~\pm~~2\sqrt{ab}}~~=~~\sqrt{a}~\pm~\sqrt{b}} \\ &=\sqrt{(18+13)+2\sqrt{18~\times~13}}~-~\sqrt{(13~+~8)-2\sqrt{13~\times~8}} \\ &=(\sqrt{18}~+~\sqrt{13}~)~-~(\sq\rt{13}-\sqrt{8})~~===\ \textgreater \ ~~\sqrt{18}+\sqrt{8}~=~
3\sqrt{2}+2\sqrt{2}~=~5\sqrt{2}
\end{align}$

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