If `a=1-sqrt(2),"then find the value of "(a-(1)/(a))^(3)`

Options

A. ` 20 sqrt(3)`
B. ` 24 sqrt(3)`
C. ` 26 sqrt(3)`
D. `22 sqrt(3)`

Answer

B

Explanation

We have, `a = 2 + sqrt(3)`
Then ` (1)/(a) = (1)/( 2 + sqrt(3)) xx (2 + sqrt(3))/(2 - sqrt(3))`
[Multiplying and dividing by ` 2 - sqrt(3)` ]
` = ( 2 - sqrt(3))/( (2)^(2) - ( sqrt(3))^(2)) = (2 - sqrt(3))/( 4 - 3) = 2 - sqrt(3)`
`rArr (1)/(a) = 2 - sqrt(3) ` [`:. (a + b) (a - b) = a^(2) - b^(2)` ]
Now, ` a - (1)/(a) = 2 + sqrt(3) - (2 - sqrt(3))`
`= 2 + sqrt(3) - 2 + sqrt(3) = 2 sqrt(3)`
Then , ` (a - (1)/(a))^(3) = ( 2 sqrt(3))^(3) = 2^(3) ( sqrt(3))^(3)`
` = 8 xx 3 sqrt(3)`
`rArr (a - (1)/(a))^(3) = 24 sqrt(3)`