Evaluate x3 +1/x3 if x=9-4square root 5
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x = 9 - 4√5
1 / x = 1 / ( 9 - 4√5 ) × ( 9 + 4√5 )/( 9 + 4√5 )
1 / x = ( 9 + 4√5 )/( 81 - 80 ) = 9 + 4√5
x³ + 1 / x³
= ( 9 - 4√5 )³ + ( 9 + 4√5 )³
[ a³ - b³ ] = a³ - b³ - 3a²b + 3ab²
[ a³ + b³ ] = a³ + b³ + 3a²b + 3ab²
= 9³ - ( 4√5 )³ - 3( 9 )²( 4√5 ) + 3( 9 )( 4√5 )² + 9³ + ( 4√5 )³ + 3( 9 )²( 4√5 ) + 3( 9 )( 4√5 )²
= 729 - 320√5 - 972√5 + 2160 + 729 + 320√5 + 972√5 + 2160
= 5778
нσρє ιт нєℓρѕ
x = 9 - 4√5
1 / x = 1 / ( 9 - 4√5 ) × ( 9 + 4√5 )/( 9 + 4√5 )
1 / x = ( 9 + 4√5 )/( 81 - 80 ) = 9 + 4√5
x³ + 1 / x³
= ( 9 - 4√5 )³ + ( 9 + 4√5 )³
[ a³ - b³ ] = a³ - b³ - 3a²b + 3ab²
[ a³ + b³ ] = a³ + b³ + 3a²b + 3ab²
= 9³ - ( 4√5 )³ - 3( 9 )²( 4√5 ) + 3( 9 )( 4√5 )² + 9³ + ( 4√5 )³ + 3( 9 )²( 4√5 ) + 3( 9 )( 4√5 )²
= 729 - 320√5 - 972√5 + 2160 + 729 + 320√5 + 972√5 + 2160
= 5778
нσρє ιт нєℓρѕ
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