Question

$1 + x = 2 + \sqrt{3} i \\ find \: \: {x}^{3} + \frac{1}{ {x}^{3} }$

Answer 1

Hey Mate !

Here is your solution :

Given,

=> x = 2 + √3 --------- ( 1 )

Multiplying by 6 and diving by 6,

=> x = [ 6( 2 +√3 ) / 6 ]

=> x = [ ( 12 + 6√3 ) / 6 ]

=> x = [ ( 9 + 3 + 6√3 ) / 6 ]

=> x = [ ( 3 )² + ( √3 )² + 2 × 3 × √3 ] / 6

Using identity :

[ a² + b² + 2ab = ( a + b )² ]

=> x = [ 3 +√3 ]² ÷ 6

=> x = ( 3 + √3 )² ÷ ( √6 )²

Using identity :

=> a^m ÷ b^m = ( a/b )^m

=> x = [ ( 3 + √3 ) / √6 ]²

=> √x = [ ( 3 + √3 ) / √6 ] ------- ( 2 )

Now,

= √x + ( 1/√x )

= [ ( √x )² + 1 ] / √x

= [ x + 1 ] / √x

Plug the value of ( 1 ) and ( 2 ) ,

= [ 2 + √3 + 1 ] / [ ( 3 + √3 ) / √6 ]

= ( 3 + √3 ) ÷ [ ( 3 + √3 ) / √6 ]

= [ ( 3 + √3 ) × √6 ] ÷ ( 3 + √3 )

= √6

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Hope it helps !! ^_^