Question

pls answer this question asap.​

Answer 1

Given Question :-

The value of

$\rm \: \: \sqrt{7 - 3 \sqrt{5} } \: is \: equals \: to$

$\: \: \: \sf \: (a) \: \: \sqrt{7} - 2 \sqrt{3}$

$\: \: \: \sf \: (b) \: \: \dfrac{3 - \sqrt{5} }{ \sqrt{2} }$

$\: \: \: \sf \: (c) \: \: \dfrac{ \sqrt{3} - \sqrt{7} }{ 2\sqrt{2} }$

$\: \: \: \sf \: (d) \: \: \dfrac{ \sqrt{5} + \sqrt{2} }{ 3 }$

$\green{\large\underline{\sf{Solution-}}}$

Given expression is

$\rm :\longmapsto\: \sqrt{7 - 3 \sqrt{5} }$

can be rewritten as

$\rm \: = \: \sqrt{\dfrac{2(7 - 3 \sqrt{5} )}{2} }$

$\rm \: = \: \sqrt{\dfrac{14 - 6 \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{9 + 5- 2 \times 3 \times \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{ {3}^{2} + { (\sqrt{5} ) \: }^{2} - 2 \times 3 \times \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{ {3}^{2} + { (\sqrt{5} ) \: }^{2} - 2 \times 3 \times \sqrt{5}}{ {( \sqrt{2}) \: }^{2} } }$

$\rm \: = \: \sqrt{\dfrac{ {3}^{2} }{ {( \sqrt{2}) \: }^{2} } + \dfrac{ {( \sqrt{5}) \: }^{2} }{ {( \sqrt{2}) \: }^{2} } - 2 \times \dfrac{3}{ \sqrt{2} } \times \dfrac{ \sqrt{5} }{ \sqrt{2} } }$

We know,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: {x}^{2} + {y}^{2} - 2xy = {(x - y)}^{2} \: }}}$

So, using this, we have

$\rm \: = \: \sqrt{ {\bigg[\dfrac{3}{ \sqrt{2} } - \dfrac{ \sqrt{5} }{ \sqrt{2} } \bigg]}^{2} }$

$\rm \: = \: \dfrac{3}{ \sqrt{2} } - \dfrac{ \sqrt{5} }{ \sqrt{2} }$

$\rm \: = \: \dfrac{3 - \sqrt{5} }{ \sqrt{2} }$

Hence,

$\rm :\longmapsto\: \boxed{ \tt{ \: \sqrt{7 - 3 \sqrt{5} } = \frac{3 - \sqrt{5} }{ \sqrt{2} } \: }}$

Thus,

• Option (b) is correct.

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Alternative Method

$\rm :\longmapsto\: \sqrt{7 - 3 \sqrt{5} }$

can be rewritten as

$\rm \: = \: \sqrt{\dfrac{2(7 - 3 \sqrt{5} )}{2} }$

$\rm \: = \: \sqrt{\dfrac{14 - 6 \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{9 + 5- 2 \times 3 \times \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{ {3}^{2} + { (\sqrt{5} ) \: }^{2} - 2 \times 3 \times \sqrt{5}}{2} }$

$\rm \: = \: \sqrt{\dfrac{ {(3 - \sqrt{5}) \: }^{2} }{ {( \sqrt{2}) \: }^{2} } }$

$\rm \: = \: \dfrac{3 - \sqrt{5} }{ \sqrt{2} }$

Hence,

$\rm :\longmapsto\: \boxed{ \tt{ \: \sqrt{7 - 3 \sqrt{5} } = \frac{3 - \sqrt{5} }{ \sqrt{2} } \: }}$

Thus,

• Option (b) is correct