Question

if a=2+√3,then fine the value of [a-1/a]​

Answer 1

GIVEN:

• a = 2 + ✔3

TO FIND:

• The value of (a -1/a)

SOLUTION:

BY RATIONALIZING THE DENOMINATOR

$\rm \: \implies \: a \: - \frac{1}{a} = \: 2 + \sqrt{3} + \frac{1}{2 + \sqrt{3} } \\ \\ \rm \: \implies \: a \: - \frac{1}{a} \: = \frac{(2 + \sqrt{3} )(2 + \sqrt{3} )}{2 + \sqrt{3} } \\ \\ \rm \: \implies \: a \: - \frac{1}{a } = \frac{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3 - 1}{2 + \sqrt{3} } \\ \\ \rm \: \implies \: a \: - \frac{1}{a} = \: \frac{7 + 4 \sqrt{3} - 1}{2 + \sqrt{3} } \\ \\ \rm \: \implies \:a \: - \frac{1}{a} = \: \frac{6 + 4 \sqrt{3} }{2 + \sqrt{3} } \\ \\ \rm \: \implies \: a \: - \frac{1}{a} = \: \frac{(6 + 4 \sqrt{3} )(2 - \sqrt{3} )}{(2 + \sqrt{3} )(2 - \sqrt{3} )} \\ \\ \rm \: \implies \: a \: - \frac{1}{a} = \: \frac{12 - 6 \sqrt{3} + 8 \sqrt{3} - 12}{(2) ^{2} - \sqrt{3} )^{2} } \\ \\ \rm \: \implies \: a \: - \frac{1}{a} = \: \frac{2 \sqrt{3} }{4 - 3} \\ \\ \rm \: \implies \: a \: - \frac{1}{a} = \: 2 + \sqrt{3}$

Therefore, Value of( a-1/a) = 2 ✔3