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Simplify, $\frac{{\sqrt {15} + \sqrt {35} + \sqrt {21} + 5}}{{\sqrt 3 + 2\sqrt 5 + \sqrt 7 }}$

 The given expression can be rearranged as $ \frac{{(\sqrt {15}  + 5) + (\sqrt {35}  + \sqrt {21} )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}$

$ = \frac{{\sqrt 5 (\sqrt 3  + \sqrt 5 ) + \sqrt 7 (\sqrt 5  + \sqrt 3 )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}$

$ = \frac{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}{{\sqrt 3  + 2\sqrt 5  + \sqrt 7 }}= \frac{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}{{(\sqrt 3  + \sqrt 5 ) + (\sqrt 5  + \sqrt 7 )}}$

$ = \frac{1}{{\frac{{(\sqrt 3  + \sqrt 5 )}}{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}} + \frac{{(\sqrt 5  + \sqrt 7 )}}{{(\sqrt 3  + \sqrt 5 )(\sqrt 5  + \sqrt 7 )}}}}$

$ = \frac{1}{{\frac{1}{{(\sqrt 5  + \sqrt 7 )}} + \frac{1}{{(\sqrt 3  + \sqrt 5 )}}}}$

$ = \frac{1}{{\frac{{(\sqrt 7  - \sqrt 5 )}}{{(\sqrt 7  + \sqrt 5 )(\sqrt 7  - \sqrt 5 )}} + \frac{{(\sqrt 5  - \sqrt 3 )}}{{(\sqrt 5  + \sqrt 3 )(\sqrt 5  - \sqrt 3 )}}}}$

$ = \frac{1}{{\frac{{(\sqrt 7  - \sqrt 5 )}}{2} + \frac{{(\sqrt 5  - \sqrt 3 )}}{2}}}$

$ = \frac{2}{{(\sqrt 7  - \sqrt 3 )}} = \frac{{2(\sqrt 7  + \sqrt 3 )}}{{(\sqrt 7  - \sqrt 3 )(\sqrt 7  + \sqrt 3 )}} = \frac{{\sqrt 7  + \sqrt 3}}{2}$

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