Question

if x=2+root3 find x^3+1/x^3

Answer 1

Hey friend,
Here is the answer you were looking for:
Given,
x = 2 + √3

Then,
1/x = 1/ 2 +√3

On rationalizing,
We get the value
1/x = 2 - √3

Now,
x^3 + (1/x)^3

= ( 2 + √3)^3 + (2 - √3)^3

= [ (2)^3 + (√3)^3 + 3 × 2 × √3 ( 2 + √3) ] + [ (2)^3 - (√3)^3 - 3 × 2 × √3 ( 2 - √3) ]

= ( 8 + 3√3 + 6√3 × 2 + 6√3 × √3 ) + ( 8 - 3√3 - 6√3 × 2 + 6√3 × √3 )

= ( 8 + 3√3 + 12√3 + 18 ) + ( 8 - 3√3 - 12√3 + 18 )

= 26 + 15√3 + 26 - 15√3

= 52

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