Question

evaluate (-1+root(-3))^9+(-1-root(-3))^9

Answer 1

{ -1 + √(-3)}^9 + { -1 - √(-3) }^9

this is based on concept of complex number

( -1 + √3i ) = 2{ -1/2 + √3/2 i }
= 2{ cos(2π/3) + sin2π/3 i }

similarly ,

( -1 - √3i ) = 2{ cos2π/3 -sin2π/3i }

now ,

{ -1 + √(-3)}^9 + { -1-√(-3)}^9 =2^9 {(cos2π/3 +sin2π/3i)}^9 +2^9 {(cos2π/3 -sin2π/3i )}^9

use De-Moivre theorem,

=2^9{( cos2π/3 ×9+sin2π/3× 9i )} + 2^9{ 2(cos2π/3×9 -9sin2π/3× 9 )}

=2^9 ( 2cos6π)

= 512 ×2 × 1 = 1024