Question

Simplify the following

Answer 1

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$\huge\sf\green{Answer}$

$\: \boxed { \rm \pink{simplify \: \: the \: \: following}}$

$\implies \bf \red{ \frac{1}{1 + \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } }$

$\: \dashrightarrow \bf{ \frac{ \sqrt{2 } + \sqrt{3} + 1 + \sqrt{2} }{1( \sqrt{2} + \sqrt{3} ) + \sqrt{2} ( \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } }$

$\: \dashrightarrow \bf{ \frac{2 \sqrt{2} + \sqrt{3} + 1}{ \sqrt{2} + \sqrt{3} + 2 + \sqrt{6} } + \frac{1}{ \sqrt{3} + \sqrt{4} } }$

$\: \dashrightarrow \bf{ \frac{ \sqrt{3}( \sqrt{2} + \sqrt{3} + 1 + \sqrt{2} ) + \sqrt{4} ( \sqrt{2} + \sqrt{3} + 1 + \sqrt{2} ) + \sqrt{2} + \sqrt{3} + 2 + \sqrt{6} }{ \sqrt{3} ( \sqrt{2} + \sqrt{3} + 2 + \sqrt{6} ) + \sqrt{4}( \sqrt{2} + \sqrt{3} + 2 + \sqrt{6} } }$

$\dashrightarrow \bf{ \frac{ \sqrt{6} + 3 + \sqrt{3} + \sqrt{6} + \sqrt{8} + \sqrt{12} + \sqrt{4} + \sqrt{8} + \sqrt{2} + \sqrt{3} + 2 \sqrt{2} + \sqrt{12} }{ \sqrt{6} + 3 + 2 \sqrt{3} + \sqrt{18} + \sqrt{8} + \sqrt{12} + 2 \sqrt{4} + \sqrt{24} } }$

$\dashrightarrow \bf{ \frac{2 \sqrt{6} + 3 + \sqrt{3} + 2 \sqrt{2} + 2 \sqrt{3} + 2 + 2 \sqrt{2} + \sqrt{2} + \sqrt{3} + 2 \sqrt{2} + 2 \sqrt{3} }{ \sqrt{6} + 3 + 2 \sqrt{3} + 3 \sqrt{2} + 2 \sqrt{2} + 2 \sqrt{3} + 2 \times 2 + 2 \sqrt{6} } }$

$\dashrightarrow \bf{ \frac{2 \sqrt{6} + 5 + 3 \sqrt{3} + 5 \sqrt{2} + 3 \sqrt{3} + 2 \sqrt{2} }{ \sqrt{6} + 7 + 4 \sqrt{3} + 5 \sqrt{2} + 2 \sqrt{6}} }$

$\pink \bigstar \boxed { \bf \purple{ \implies\frac{2 \sqrt{ 6} + 5 + 7\sqrt2+ 6 \sqrt{3} }{ 3 \sqrt{6} + 7 + 4 \sqrt{3} + 5 \sqrt{2} } }}$