Given :
[tex] \\ \\ [/tex]
To find : Rationalize the Denominator
[tex] \\ \qquad{\rule{200pt}{2pt}} [/tex]
SolutioN :
[tex] \dag \; {\underline{\underline{\textsf{\textbf{ \; Rationalizing \; :- }}}}} [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ \sqrt{3} - 1 }{ \sqrt{3} + 1 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} - 1 ) \times ( \sqrt{3} - 1 ) }{ ( \sqrt{3} + 1 ) \times ( \sqrt{3} - 1 ) } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} - 1 )^2 }{ ( \sqrt{3} )^2 - ( 1 )^2 } } \\ \\ [/tex]
[tex] \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} )^2 + (1)^2 + 2( \sqrt{3} )(1) }{ ( \sqrt{3} )^2 - ( 1 )^2 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( 3 + 1 ) + 2 \sqrt{3} }{ 3 - 1 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ 4 + 2 \sqrt{3} }{ 2 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{4}{2} + \dfrac{2 \sqrt{3}}{2} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \cancel\dfrac{4}{2} + \dfrac{ \cancel2 \sqrt{3}}{ \cancel2} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { 2 + \sqrt{3} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; {\pmb{\underline{\boxed{\purple{\frak { \dfrac{ \sqrt{3} - 1 }{ \sqrt{3} + 1 } = 2 + \sqrt{3} }}}}}} \; \bigstar \\ \\ [/tex]
[tex] \therefore \; [/tex] The Required Answer is 2 + √3
[tex] \\ {\underline{\rule{300pt}{9pt}}} [/tex]
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Verified answer
Given :
[tex] \\ \\ [/tex]
To find : Rationalize the Denominator
[tex] \\ \qquad{\rule{200pt}{2pt}} [/tex]
SolutioN :
[tex] \dag \; {\underline{\underline{\textsf{\textbf{ \; Rationalizing \; :- }}}}} [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ \sqrt{3} - 1 }{ \sqrt{3} + 1 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} - 1 ) \times ( \sqrt{3} - 1 ) }{ ( \sqrt{3} + 1 ) \times ( \sqrt{3} - 1 ) } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} - 1 )^2 }{ ( \sqrt{3} )^2 - ( 1 )^2 } } \\ \\ [/tex]
[tex] \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( \sqrt{3} )^2 + (1)^2 + 2( \sqrt{3} )(1) }{ ( \sqrt{3} )^2 - ( 1 )^2 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ ( 3 + 1 ) + 2 \sqrt{3} }{ 3 - 1 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{ 4 + 2 \sqrt{3} }{ 2 } } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \dfrac{4}{2} + \dfrac{2 \sqrt{3}}{2} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { \cancel\dfrac{4}{2} + \dfrac{ \cancel2 \sqrt{3}}{ \cancel2} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; \sf { 2 + \sqrt{3} } \\ \\ [/tex]
[tex] \; \; \; :\implies \; \; {\pmb{\underline{\boxed{\purple{\frak { \dfrac{ \sqrt{3} - 1 }{ \sqrt{3} + 1 } = 2 + \sqrt{3} }}}}}} \; \bigstar \\ \\ [/tex]
[tex] \\ \\ [/tex]
[tex] \therefore \; [/tex] The Required Answer is 2 + √3
[tex] \\ {\underline{\rule{300pt}{9pt}}} [/tex]