Question

$\sqrt{12 - 2 \sqrt{35} }$
Plz help me.....​

Answer 1

Answer:

$12 + 2 \sqrt{35} \\ = ( \sqrt{7 } + \sqrt{ {5)}^{2} } \\ = \sqrt{12 + 2 \sqrt{35} } \\ = \sqrt{7} + \sqrt{5}$

Answer 2

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

Given expression is

$\sf \: \sqrt{12 - 2 \sqrt{35} }$

$\sf \: = \: \sqrt{7 + 5- 2 \sqrt{35} }$

$\sf \: = \: \sqrt{ {( \sqrt{7}) }^{2} + {( \sqrt{5} )}^{2} - 2 \sqrt{7 \times 5} }$

$\sf \: = \: \sqrt{ {( \sqrt{7}) }^{2} + {( \sqrt{5} )}^{2} - 2 \sqrt{7} \sqrt{5} }$

We know,

$\boxed{ \sf{ \: {x}^{2} + {y}^{2} - 2xy = {(x - y)}^{2} \: }}$

So, here

$\sf \: x = \sqrt{7}$

$\sf \: y = \sqrt{5}$

So, using above identity, we get

$\sf \: = \: \sqrt{ {( \sqrt{7} - \sqrt{5}) }^{2} }$

$\sf \: = \: \sqrt{7} - \sqrt{5}$

Hence,

$\sf \:\bf\implies \: \sqrt{12 - 2 \sqrt{35} } = \: \sqrt{7} - \sqrt{5}$

$\rule{190pt}{2pt}$

Additional Information

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$