Question

√(√3-√(4-√5-√(17-4√15)))

Answer 1

I think you question is $\sqrt{\sqrt{3}-\sqrt{4-\sqrt{5}-\sqrt{17-4\sqrt{15}}}}$

first of all, solve √(17 - 4√15) ,
√(17 - 4√15) = √(√12² + √5² - 2√12.√5)
= √(√12 - √5)²
= (√12 - √5) = (2√3 - √5)

so put (2√3 - √5) in place of √(17 - 4√15)

now, $\sqrt{\sqrt{3}-\sqrt{4-\sqrt{5}-2\sqrt{3}+\sqrt{5}}}$

= $\sqrt{\sqrt{3}-\sqrt{4-2\sqrt{3}}}$

√(4 - 2√3) = √{√3² + 1² - 2√3} = √3 - 1

= $\sqrt{\sqrt{3}-\sqrt{3}+1}$

= 1

Answer 2

HELLO DEAR,

GIVEN:-
$\bold{\sqrt{ \sqrt{3} - \sqrt{4 - \sqrt{5} - \sqrt{17 - 4 \sqrt{5} } } } }$

$\bold{\implies \sqrt{\sqrt{3} - \sqrt{4 - \sqrt{5} - \sqrt{12 + 5 - 2.\sqrt{12}.\sqrt{5}}}}}$

$\bold{\implies \sqrt{\sqrt{3} - \sqrt{4 - \sqrt{5} - \sqrt{(\sqrt{12} - \sqrt{5})^2}}}}$

$\bold{\implies \sqrt{\sqrt{3} - \sqrt{4 - \sqrt{5} - \sqrt{12} + \sqrt{5}}}}$

$\bold{\implies \sqrt{\sqrt{3} - \sqrt{3 + 1 - 2.1.\sqrt{3}}}}$

$\bold{\implies \sqrt{\sqrt{3} - \sqrt{(\sqrt{3} - 1)^2}}}$

$\bold{\implies \sqrt{\sqrt{3} + 1 - \sqrt{3}}}$

$\bold{\implies 1}$

I HOPE IT'S HELP YOU DEAR,
THANKS