Question

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

Answer 1

Question :

If a=3-√2 Find the value of $a-\dfrac{1}{a}$

Solution :

Given : a= 3-√2 .....(1)

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

Rationalise the denominator

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

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

we know that $a{}^{2}-b{}^{2}=(a+b)(a-b)$

$\implies \dfrac{1}{a} = \dfrac{3 + \sqrt{2} }{3 {}^{2} - ( \sqrt{2} ) {}^{2} }$

$\implies \dfrac{1}{a} = \dfrac{3 + \sqrt{2} }{9 - 2}$

$\implies \dfrac{1}{a} = \dfrac{3 + \sqrt{2} }{7} ....(2)$

___________________________

we have to find the value of $a-\dfrac{1}{a}$

put the equation (1)&(2) values

$\implies (a - \frac{1}{a} ) = (3 - \sqrt{2} - \frac{3 + \sqrt{2} }{7} )$

$= \dfrac{21 - 7 \sqrt{2} - 3 + \sqrt{2} }{7}$

$= \dfrac{(21 - 3) - \sqrt{2}(7 - 1) }{7}$

$= \dfrac{18 - 6 \sqrt{2} }{7}$

it is the required solution!