Let = -1+i 3 2 and = -1-i 3 2 , i= -1 . If (7-7 +9 )²⁰+(9+7 -7 )²⁰+(-7+9 +7 )²⁰+(14+7 +7 )²⁰=m¹⁰ , then m is _ _ _ _ _

Step-by-step solution

= , = ^2 (cube roots of unity). Let z₁ = 7-7 +9 ^2 , z₂ = 9+7 -7 ^2 , z₃ = -7+9 +7 ^2 . Calculating: z₁ = 6-8i 3 , z₂ = 9+7i 3 , z₃ = -15+i 3 . All have |z_k| = 228 . Verify: z₂ = z₁ , z₃ = z₁ ^2 . z₁²⁰+z₂²⁰+z₃²⁰ = z₁²⁰(1+ ²⁰+ ⁴⁰) = z₁²⁰(1+ ^2+ ) = 0 . Fourth term: 14+7 +7 ^2 = 14+7(-1) = 7 . So 7²⁰ = 49¹⁰ . Total = 0 + 49¹⁰ = m¹⁰ m = 49 .