Question

If x= 9-4 root 5 find x^3 +1/x^3

Answer 1

x = 9 - 4√5 ----( 1 )

1/x = 1 / ( 9 - 4√5 )

= ( 9 + 4√5 ) / [ ( 9 - 4√5 ) ( 9 + 4√5 )

= ( 9 + 4√5 ) / [ 9² - ( 4√5 )² ]

= ( 9 + 4√5 ) / [ 81 - 80 ]

= 9 + 4√5 ---( 2 )

x - 1/x = 9 - 4√5 - ( 9 + 4√5 )

= 9 - 4√5 - 9 - 4√5

= -8√5 ---( 3 )

x³ - 1/x³ = ( x - 1/x )³ - 3( x - 1/x )

= ( - 8√5 )³ - 3 × ( - 8√5 )

= ( -8√5 ) [ ( - 8√5 )² - 3 ]

= ( -8√5 ) [ 320 - 3 ]

= ( -8√5 ) × 317

= - 2536√5

I hope this helps you.