Given: #160;a=2+3Now, #160;1a=12+3Multiply #160;and #160;divide #160;RHS #160;by #160;2 3, #160;we #160;get #8658;1a=12+3 #215;2 32 3 #8 ...

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Finding the Value of an Expression

If a=2+3, the...

Question

If $a = 2 + \sqrt{3}, then a - \frac{1}{a} =___________.$

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Solution

$Given : a = 2 + \sqrt{3} Now, \frac{1}{a} = \frac{1}{2 + \sqrt{3}} Multiply and divide RHS by (2 - \sqrt{3}), we get \Rightarrow \frac{1}{a} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} \Rightarrow \frac{1}{a} = \frac{2 - \sqrt{3}}{{(2)}^{2} - {(\sqrt{3})}^{2}} (Using the identity : (a + b) (a - b) = a^{2} - b^{2}) \Rightarrow \frac{1}{a} = \frac{2 - \sqrt{3}}{4 - 3} \Rightarrow \frac{1}{a} = \frac{2 - \sqrt{3}}{1} \Rightarrow \frac{1}{a} = 2 - \sqrt{3} Thus, a - \frac{1}{a} = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2 \sqrt{3}$

Hence, if $a = 2 + \sqrt{3}, then a - \frac{1}{a} = 2 \sqrt{3} .$

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