यदि `x=sqrt(3)+sqrt(2)` है तो `(x^(3)+1/(x^(3)))` का मान ज्ञात करें।

Options

A. `6sqrt(3)`
B. `12sqrt(3)`
C. `18sqrt(3)`
D. `24sqrt(3)`

Answer

C

Explanation

`x=sqrt(3)+sqrt(2)`
`:. 1/x=sqrt(3)-sqrt(2)`
`x^(3)+1/(x^(3))`
`:. x^(3)=(sqrt(3)+sqrt(2))^(3)`
`=(sqrt(3))^(3)+(sqrt(2))^(3)+3xxsqrt(3)xxsqrt(2)(sqrt(3)+sqrt(2))`
`=3sqrt(2)+2sqrt(2)+3sqrt(6)(sqrt(3)+sqrt(2))`
`=3sqrt(3)+2sqrt(2)+9sqrt(2)+6sqrt(3)`
`x^(3)=9sqrt(3)+11sqrt(2)`
`1/(x^(3))=9sqrt(3)-11sqrt(2)`
`x^(3)+1/(x^(3))=9sqrt(3)+11sqrt(2)+9sqrt(3)-11sqrt(2)`
`=18sqrt(3)`
Alternate:
`x=sqrt(3)+sqrt(2)`
`1/x=sqrt(3)-sqrt(2)`
and `x+1/x=sqrt(3)+sqrt(2)+sqrt(3)-sqrt(2)`
`=2sqrt(3)`
`x^(3)+1/(x^(3))=(x+1/x)^(3)-3(x+1/x)`
`=(2sqrt(3))^(3)-3(2sqrt(3))`
`=24sqrt(3)-6sqrt(3)=18sqrt(3)`