Question

$\rm{Find \: the \: value \: of :}$
$\rm{ \sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } }$

Answer 1

$\underline{ \underline{ \sf{ \maltese \:Question }}} :$

$\rm{Find \: the \: value \: of :}\rm{ \sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } }$

$\underline{ \underline{ \sf{ \maltese \:Answer }}} :$

$\\ \mathtt{ Let : \: \: } \sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } = \rm{\sqrt{x} + \sqrt{y} - \sqrt{z} }$

$\mathrm \red{S.O.B.S}$

$\\ \sf{ \sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } = x + y + z + 2 \sqrt{xy} - 2 \sqrt{yz} - 2 \sqrt{zx} }$

$\\ \rm \red{By \: comparing \: we \: get ;}$

$\\ \sf{x + y + z = 9 \: ; \: xy = 15 \: ; \: yz \: = \: 5 \: ; \: zx \: = \: 3}$

$\\ \rm \red{Now : }$

$\\ \sf{(xy)(yz)(zx) = (15)(5)(3)}$

$\sf{ {x}^{2} {y}^{2} {z}^{2} = 225}$

$\sf{xyz = \sqrt{225} }$

$\boxed{ \sf{xyz = 15}}$

Case i :-

$\sf{xyz = 15 \: \implies \: x = \frac{15}{yz} \implies \: x = \frac{15}{5} = 3}$

Case ii :-

$\sf{xyz = 15 \: \implies \: y = \frac{15}{xz} \implies \: y = \frac{15}{3} = 5}$

Case iii :-

$\sf{xyz = 15 \: \implies \: z = \frac{15}{xy} \implies \: z = \frac{15}{15} = 1}$

$\rm \red{From \: above \: cases }$

$\sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } = \rm{\sqrt{3} + \sqrt{5} - {1} }$

Answer 2

$\rm{Find \: the \: value \: of :}$

$\rm{ \sqrt{9 - 2 \sqrt{3} - 2 \sqrt{5} + 2 \sqrt{15} } }$