Question

VA ROG FRUMOS MA PUTETI AJUTA SI PE MINE AICI

Answer 1

 

$\displaystyle\\ d)\\\frac{2}{\sqrt{16}-\sqrt{14}}-\frac{2}{\sqrt{14}-\sqrt{12}}+\frac{2}{\sqrt{12}-\sqrt{10}}-\frac{2}{\sqrt{10}-\sqrt{8}}+\frac{2}{\sqrt{8}-\sqrt{6}}-\frac{2}{\sqrt{6}-\sqrt{4}}\\\\\text{Rationalizam numitorii si la fiecare fractie vom avea la numitor numarul 2}\\\text{care se simplifica cu 2 de la numarator.}\\\\\text{Expresia devine:}\\\\\sqrt{16}+\sqrt{14}-(\sqrt{14}+\sqrt{12})+\sqrt{12}+\sqrt{10}-(\sqrt{10}+\sqrt{8})+\sqrt{8}+\sqrt{6}-(\sqrt{6}+\sqrt{4})$

$\displaystyle\\\sqrt{16}+\sqrt{14}-(\sqrt{14}+\sqrt{12})+\sqrt{12}+\sqrt{10}-(\sqrt{10}+\sqrt{8})+\sqrt{8}+\sqrt{6}-(\sqrt{6}+\sqrt{4})\\\\\text{Eliminam parantezele.}\\\\\sqrt{16}+\sqrt{14}-\sqrt{14}-\sqrt{12}+\sqrt{12}+\sqrt{10}-\sqrt{10}-\sqrt{8})+\sqrt{8} +\sqrt{6}-\sqrt{6}-\sqrt{4}\\\\\text{Reducem termenii asemenea si ramanem cu:}\\\\\sqrt{16}-\sqrt{4} = 4-2=\boxed{\bf2}$

$\displaystyle\\e)\\\frac{1}{\sqrt{9}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{4}}=\\\\\\=\frac{\sqrt{9}-\sqrt{8}}{9-8}+\frac{\sqrt{8}-\sqrt{7}}{8-7}+\frac{\sqrt{7}-\sqrt{6}}{7-6}+\frac{\sqrt{6}-\sqrt{5}}{6-5}+\frac{\sqrt{5}-\sqrt{4}}{5-4}=$

$\displaystyle\\=\frac{\sqrt{9}-\sqrt{8}}{1}+\frac{\sqrt{8}-\sqrt{7}}{1}+\frac{\sqrt{7}-\sqrt{6}}{1}+\frac{\sqrt{6}-\sqrt{5}}{1}+\frac{\sqrt{5}-\sqrt{4}}{1}=\\\\=\sqrt{9}-\sqrt{8}+\sqrt{8}-\sqrt{7}+\sqrt{7}-\sqrt{6}+\sqrt{6}-\sqrt{5}+\sqrt{5}-\sqrt{4}=\\\\=\sqrt{9}-\sqrt{4}=3-2=\boxed{1}$

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