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Answer:
[tex] = { \tt{ \frac{1}{( \sqrt{4} - \sqrt{2} )}. \frac{ \sqrt{4} + \sqrt{2} }{ \sqrt{4} + \sqrt{2} } }} \\ \\ = { \tt{ \frac{ \sqrt{4} + \sqrt{2} }{( \sqrt{4}) {}^{2} - {( \sqrt{2}) }^{2} } }} \\ \\ = { \tt{ \frac{ \sqrt{4} + \sqrt{2} }{4 - 2} }} \\ \\ = { \tt{ \frac{2 + \sqrt{2} }{2} } } \\ \\ = { \boxed{ \tt{ \: \: 1 + \frac{ \sqrt{2} }{2} }}}[/tex]
Step-by-step explanation:
[tex]\underline{\underline{\sf{➤\:\: Solution }}}[/tex]
[tex] \sf(a) \: \: \: \dfrac{1}{ \sqrt{4} - \sqrt{2} } [/tex]
On rationalising,
[tex] \sf \implies \dfrac{1}{ \sqrt{4} - \sqrt{2} } \times \dfrac{\sqrt{4} + \sqrt{2} }{\sqrt{4} + \sqrt{2} } [/tex]
Combine the fractions,
[tex] \sf \implies \dfrac{1(\sqrt{4} + \sqrt{2}) }{(\sqrt{4} - \sqrt{2})(\sqrt{4} + \sqrt{2}) } [/tex]
We know that,
[tex] \sf \implies (a - b)(a + b) = (a)^{2} - (b)^{2} [/tex]
So,
[tex] \sf \implies \dfrac{1(\sqrt{4} + \sqrt{2}) }{(\sqrt{4})^{2} - (\sqrt{2}) ^{2} }[/tex]
[tex] \sf \implies \dfrac{1(\sqrt{4} + \sqrt{2}) }{4 - 2 }[/tex]
[tex] \sf \implies \dfrac{1(\sqrt{4} + \sqrt{2}) }{2 }[/tex]
[tex] \sf \implies \dfrac{\sqrt{4} + \sqrt{2}}{2} [/tex]
Hence,
Hence, On rationalising we got,
[tex]\implies \bf {\dfrac{\sqrt{4} + \sqrt{2}}{2}} [/tex]