[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: \dfrac{12}{ \sqrt{10} + \sqrt{7} + \sqrt{3} } \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{12}{ (\sqrt{7} + \sqrt{3}) + \sqrt{10} } \\ \\ [/tex]
On rationalizing the denominator, we get
[tex]\sf \: = \: \dfrac{12}{ (\sqrt{7} + \sqrt{3}) + \sqrt{10} } \times \frac{( \sqrt{7} + \sqrt{3}) - \sqrt{10} }{( \sqrt{7} + \sqrt{3} ) - \sqrt{10} } \\ \\ [/tex]
We know,
[tex]\boxed{ \sf{ \:(x + y)(x - y) = {x}^{2} - {y}^{2} \: }} \\ \\ [/tex]
So, using this result, we get
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ {( \sqrt{7} + \sqrt{3}) }^{2} - {( \sqrt{10}) }^{2} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ {( \sqrt{7} )}^{2} + {( \sqrt{3} )}^{2} + 2( \sqrt{7})( \sqrt{3}) - 10 } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ 7 + 3 + 2 \sqrt{21} - 10 } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ 2 \sqrt{21} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{6( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{\sqrt{21} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{6( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{\sqrt{21} } \times \frac{ \sqrt{21} }{ \sqrt{21} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{6( \sqrt{7 \times 21} + \sqrt{3 \times 21} - \sqrt{10 \times 21} )}{21} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{2( \sqrt{7 \times 7 \times 3} + \sqrt{3 \times 3 \times 7} - \sqrt{ 210} )}{7} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{2( 7 \sqrt{3} + 3 \sqrt{7} - \sqrt{ 210} )}{7} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{14\sqrt{3} + 6 \sqrt{7} - 2\sqrt{ 210}}{7} \\ \\ [/tex]
Hence,
[tex]\sf \: \implies \: \dfrac{12}{ \sqrt{10} + \sqrt{7} + \sqrt{3} } = \: \dfrac{14\sqrt{3} + 6 \sqrt{7} - 2\sqrt{ 210}}{7} \\ \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
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Answers & Comments
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: \dfrac{12}{ \sqrt{10} + \sqrt{7} + \sqrt{3} } \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{12}{ (\sqrt{7} + \sqrt{3}) + \sqrt{10} } \\ \\ [/tex]
On rationalizing the denominator, we get
[tex]\sf \: = \: \dfrac{12}{ (\sqrt{7} + \sqrt{3}) + \sqrt{10} } \times \frac{( \sqrt{7} + \sqrt{3}) - \sqrt{10} }{( \sqrt{7} + \sqrt{3} ) - \sqrt{10} } \\ \\ [/tex]
We know,
[tex]\boxed{ \sf{ \:(x + y)(x - y) = {x}^{2} - {y}^{2} \: }} \\ \\ [/tex]
So, using this result, we get
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ {( \sqrt{7} + \sqrt{3}) }^{2} - {( \sqrt{10}) }^{2} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ {( \sqrt{7} )}^{2} + {( \sqrt{3} )}^{2} + 2( \sqrt{7})( \sqrt{3}) - 10 } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ 7 + 3 + 2 \sqrt{21} - 10 } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{12( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{ 2 \sqrt{21} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{6( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{\sqrt{21} } \\ \\ [/tex]
On rationalizing the denominator, we get
[tex]\sf \: = \: \dfrac{6( \sqrt{7} + \sqrt{3} - \sqrt{10} )}{\sqrt{21} } \times \frac{ \sqrt{21} }{ \sqrt{21} } \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{6( \sqrt{7 \times 21} + \sqrt{3 \times 21} - \sqrt{10 \times 21} )}{21} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{2( \sqrt{7 \times 7 \times 3} + \sqrt{3 \times 3 \times 7} - \sqrt{ 210} )}{7} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{2( 7 \sqrt{3} + 3 \sqrt{7} - \sqrt{ 210} )}{7} \\ \\ [/tex]
[tex]\sf \: = \: \dfrac{14\sqrt{3} + 6 \sqrt{7} - 2\sqrt{ 210}}{7} \\ \\ [/tex]
Hence,
[tex]\sf \: \implies \: \dfrac{12}{ \sqrt{10} + \sqrt{7} + \sqrt{3} } = \: \dfrac{14\sqrt{3} + 6 \sqrt{7} - 2\sqrt{ 210}}{7} \\ \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]