Question

please help me with this!​

Answer 1

$\mathrm{1.\;Question :\;\dfrac{1}{\sqrt{5} + \sqrt{3}} + \dfrac{\sqrt{5} - \sqrt{3}}{2}}$

$\mathrm{Multiply\;and\;Divide\;first\;fraction\;with\;\sqrt{5} - \sqrt{3}}$

$\mathrm{\longrightarrow \dfrac{\sqrt{5} - \sqrt{3}}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} + \dfrac{\sqrt{5} - \sqrt{3}}{2}}$

$\mathrm{\longrightarrow \dfrac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} + \dfrac{\sqrt{5} - \sqrt{3}}{2}}$

$\mathrm{\longrightarrow \dfrac{\sqrt{5} - \sqrt{3}}{5 - 3} + \dfrac{\sqrt{5} - \sqrt{3}}{2}}$

$\mathrm{\longrightarrow \dfrac{\sqrt{5} - \sqrt{3}}{2} + \dfrac{\sqrt{5} - \sqrt{3}}{2}}$

$\mathrm{\longrightarrow 2\left(\dfrac{\sqrt{5} - \sqrt{3}}{2}\right)}$

$\mathrm{\longrightarrow \sqrt{5} - \sqrt{3}}$

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$\mathrm{2.\;Consider :\;\dfrac{7 + \sqrt{3}}{7 - \sqrt{3}}}$

$\mathrm{Multiply\;and\;Divide\;with\;\sqrt{7} + \sqrt{3}}$

$\mathrm{\longrightarrow\dfrac{(7 + \sqrt{3})(7 + \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}}$

$\mathrm{\longrightarrow\dfrac{(7 + \sqrt{3})^2}{(7)^2 - (\sqrt{3})^2}}$

$\mathrm{\longrightarrow\dfrac{(7)^2 + (\sqrt{3})^2 + 2(7)(\sqrt{3})}{49 - 3}}$

$\mathrm{\longrightarrow\dfrac{49 + 3 + 14\sqrt{3}}{46}}$

$\mathrm{\longrightarrow\dfrac{52 + 14\sqrt{3}}{46}}$

$\mathrm{Consider :\;\dfrac{7 - \sqrt{3}}{7 + \sqrt{3}}}$

$\mathrm{Multiply\;and\;Divide\;with\;\sqrt{7} - \sqrt{3}}$

$\mathrm{\longrightarrow\dfrac{(7 - \sqrt{3})(7 - \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}}$

$\mathrm{\longrightarrow\dfrac{(7 - \sqrt{3})^2}{(7)^2 - (\sqrt{3})^2}}$

$\mathrm{\longrightarrow\dfrac{(7)^2 + (\sqrt{3})^2 - 2(7)(\sqrt{3})}{49 - 3}}$

$\mathrm{\longrightarrow\dfrac{49 + 3 - 14\sqrt{3}}{46}}$

$\mathrm{\longrightarrow\dfrac{52 - 14\sqrt{3}}{46}}$

$\mathrm{Question :\;\dfrac{7 + \sqrt{3}}{7 - \sqrt{3}} + \dfrac{7 - \sqrt{3}}{7 + \sqrt{3}}}$

$\mathrm{\longrightarrow\dfrac{52 + 14\sqrt{3}}{46} + \dfrac{52 - 14\sqrt{3}}{46}}$

$\mathrm{\longrightarrow\dfrac{52 + 14\sqrt{3} + 52 - 14\sqrt{3}}{46}}$

$\mathrm{\longrightarrow\dfrac{104}{46}}$

$\mathrm{\longrightarrow\dfrac{52}{23}}$

Answer 2

hope it helps

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