Question

If \[a = \sqrt{2} + 1\],then find the value of  \[a - \frac{1}{a}\].

Answer 1

Given that,`a = sqrt2` +1 hence `1/a`is given as

`1/a = 1/(sqrt2+1)`we are asked to find  `a-1/a`

We know that rationalization factor for `sqrt2 +1` is `sqrt2 -1`. We will multiply each side of the given expression  `1/(sqrt2+1)`by, `sqrt2-1` to get

`1/a = 1/(sqrt2+1) xx (sqrt2-1)/(sqrt2-1)`

` = (sqrt2 - 1) /((sqrt2^2) - (1)^2)`

`= (sqrt2 -1)/(2-1)`

`=sqrt2 - 1`

Therefore,

` a-1/a = sqrt2 +1- (sqrt2-1)`

` = sqrt2+1-sqrt2 +1`

  `= 2`

Hence value of the given expression is 2.