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Step-by-step explanation:
[tex] \sf \: \sqrt{3} = 1.732, \: then \: \sqrt{\dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1} } [/tex]
[tex] \red{\large\underline{\sf{Solution-}}}[/tex]
Given expression is
[tex]\rm :\longmapsto\: \sqrt{\dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1} } [/tex]
On rationalizing the denominator, we get
[tex]\rm \: = \: \sqrt{\dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1} \times \dfrac{ \sqrt{3} - 1}{ \sqrt{3} - 1} } [/tex]
[tex]\rm \: = \: \sqrt{\dfrac{ {( \sqrt{3} - 1)}^{2} }{( \sqrt{3} + 1)( \sqrt{3} - 1)} } [/tex]
We know,
[tex] \boxed{ \tt \: (x - y)(x + y) = {x}^{2} - {y}^{2} \: }[/tex]
So, using this, we get
[tex]\rm \: = \: \dfrac{ \sqrt{3} - 1}{ \sqrt{ {( \sqrt{3})}^{2} - {(1)}^{2} } } [/tex]
[tex]\rm \: = \: \dfrac{ \sqrt{3} - 1}{ \sqrt{ 3 - 1} } [/tex]
[tex]\rm \: = \: \dfrac{ \sqrt{3} - 1}{ \sqrt{2} } [/tex]
[tex]\rm \: = \: \dfrac{ \sqrt{3} - 1}{ \sqrt{2} } \times \dfrac{ \sqrt{2} }{ \sqrt{2} } [/tex]
[tex]\rm \: = \: \dfrac{(1.732 - 1) \sqrt{2} }{2} [/tex]
[tex]\rm \: = \: \dfrac{0.732 \times \sqrt{2} }{2} [/tex]
[tex]\rm \: = \: 0.366 \sqrt{2} [/tex]
Hence,
[tex]\rm :\longmapsto\: \boxed{ \rm{ \: \: \: \sqrt{\dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1} } = 0.366 \sqrt{2} \: \: \: }}[/tex]
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(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)² = (a - b)² + 4ab
(a - b)² = (a + b)² - 4ab
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a - b)³ = a³ - b³ - 3ab(a - b)