Question

solve and find (a+b). in above equation.​

Answer 1

a+b = 16.

$2( \sqrt{3 + \sqrt{5 - \sqrt{13 + \sqrt{48} } } } ) \\ = 2( \sqrt{3 + \sqrt{5 - \sqrt{13 + 4 \sqrt{3} } } } ) \\ = 2( \sqrt{3 + \sqrt{5 - \sqrt{1 + (2 \times 2 \sqrt{3} \times 1 )+ 12 } } } ) \\ = 2( \sqrt{3 + \sqrt{5 - \sqrt{ {(1 + 2 \sqrt{3} )}^{2} } } } ) \\ = 2( \sqrt{3 + \sqrt{5 - 1 - 2 \sqrt{3} } } ) \\ = 2( \sqrt{3 + \sqrt{4 - 2 \sqrt{3} } } ) \\ = 2( \sqrt{1 + \sqrt{1 - (2 \times \sqrt{3} \times 1) + 3} } ) \\ = 2( \sqrt{3 + \sqrt{ {( 1-\sqrt{3} )}^{2} } } \\ = 2( \sqrt{3 + 1-\sqrt{3} } ) \\ = 2( \sqrt{4+ \sqrt{3} } ) = \sqrt{a} + \sqrt{b}$

Squaring both side we get,

$4 {( \sqrt{4+ \sqrt{3} } )}^{2} = {( \sqrt{a} + \sqrt{b} )}^{2} \\ 4(4+ \sqrt{3} ) = a + b + 2 \sqrt{ab} \\ 16 + 4 \sqrt{3} = a + b + 2 \sqrt{ab}$

Thus, a+b = 16