Question

if a=2+√3 then find the value of a-1/a​

Answer 1

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If a=2+√3 then find the value of $a- \frac{1}{a}$

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• a = 2 + √3

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• Find the value of $a - \frac{1}{a}$

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$\frac{1}{a} = \frac{1}{2 + \sqrt{3}}$

On the rationalizing this denominator we get,

$\frac{1}{a} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$

Using the identity:

★ $(x+y)(x-y) = {x}^{2}-{y}^{2}$ ★

$\frac{1}{a} = \frac{2 - \sqrt{3} }{ {(2)}^{2} - { (\sqrt{3}) }^{2} }$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3}$

$\frac{1}{a} = \frac{2 - \sqrt{3} }{1}$

$\frac{1}{a} = 2 - \sqrt{3}$

$a - \frac{1}{a}$

Now,

Putting the values,

$a - \frac{1}{a} = (2 + \sqrt{3} ) - (2 - \sqrt{3} )$

$a - \frac{1}{a} = \cancel{2} + \sqrt{3} \cancel{- 2} + \sqrt{3}$

$a - \frac{1}{a} = \sqrt{3} + \sqrt{3}$

$a - \frac{1}{a} = 2 \sqrt{3}$

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