Question

PLEASEEEE If z=-1-sqrt3i and z^6 = a+bi, then a=___ and b=___.

Answer 1

a = 94

b = -66√3

Explanation:

Given z =
`-1-i√(3)`

`z^(6)=(z^(3))^(2)`

`(-1-i√(3))^6=[(-1-i√(3))^3]^2`

`=[-(1+i√(3))^3]^(2)`

`=[(1+i√(3))^3]^2`

`=[(1)^3+(i√(3))^3+3(1)^2(i√(3))+3(1)(i√(3))^2]^2` [(a + b)³ = a³ + b³ + 3a²b + 3ab²]

= [1 + 3i² + 3i√3 + 9i²]²

= [1 - 3 + 3i√3 - 9]²

= [-11 + 3i√3]²

= (11)² -2(3i√3)(11) + (3i√3)²

= 121 - 66i√3 + 27i²

= 121 - 66i√3 - 27

= (94 - 66i√3)

Comparing it with (a + bi),

a = 94 and b = -66√3