Question

If a=2+√3, then find value of √a-√(1/a)

Answer 1

2√3 is the answer of this question

Answer 2

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \:If \: a = 2 + \sqrt{3} \: then \: find \: \sqrt{a} - \dfrac{1}{ \sqrt{a} }$

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$\bf \:\huge\red{AηsωeR : } ✍$

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$\bf \:❥︎ \: Step :- 1.$

$\sf \:  ⟼a = 2 + \sqrt{3}$

$\sf \:  ⟼\dfrac{1}{ a} = \dfrac{1}{2 + \sqrt{3} }$

❥︎ On rationalizing the denominator, we get

$\sf \:  ⟼\dfrac{1}{ a} = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\sf \:  ⟼\dfrac{1}{ a} = \dfrac{ 2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3}) }^{2} }$

$\sf \:  ⟼\dfrac{1}{ a} = \dfrac{2 - \sqrt{3} }{4 - 3}$

$\bf\implies \: \dfrac{1}{ a} = 2 - \sqrt{3}$

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$\bf \:❥︎ \: Step :- 2.$

$\bf \:  ⟼ Consider \: { \bigg( \sqrt{a} - \dfrac{1}{ \sqrt{a} } \bigg )}^{2}$

$\sf \:  ⟼ \: a + \dfrac{1}{a} - 2 \times \cancel{\sqrt{ a} } \times \dfrac{1}{ \cancel{\sqrt{a} }}$

$\bf \:  ⟼ a + \dfrac{1}{a} - 2$

$\bf \:  ⟼ 2 + \sqrt{3} + 2 - \sqrt{3} - 2$

$\bf \:  ⟼ 2 + \cancel{\sqrt{3}} + 2 - \cancel{\sqrt{3} } - \cancel{2}$

$\bf\implies \:{ \bigg( \sqrt{a} - \dfrac{1}{ \sqrt{a} } \bigg )}^{2} = 2$

$\bf\implies \:{ \bigg( \sqrt{a} - \dfrac{1}{ \sqrt{a} } \bigg )} = \sqrt{2}$

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$\large{\boxed{\boxed{\bf{Hence, \: \sqrt{a} - \dfrac{1}{ \sqrt{a} } = \sqrt{2} }}}}$

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