Question

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

`(-2sqrt(3) - 2"i")^5`

Answer 1

Let z = `-2sqrt(3) - 2"i"`

∴ a = `-2sqrt(3)`, b = –2, i.e. a < 0, b < 0

∴ |z| = `sqrt("a"^2 + "b"^2) = sqrt((-2sqrt(3))^2 + (-2)^2) = sqrt(12 + 4)` = 4

Here `(-2sqrt(3), -2)` lies in 3^rd quadrant.

∴ amp (z) = `pi + tan^-1("b"/"a")`

= `pi + tan^-1((-2)/(-2sqrt(3)))`

= `pi - tan^-1(1/sqrt(3))`

= `pi - pi/6`

= `(5pi)/6`

z⁵ = `(-2sqrt(3) - 2"i")^5`

= `[4(cos  (5pi)/6 + "i"sin  (5pi)/6)]^5`

= `1024(cos  (25pi)/6 + "i"sin  (25pi)/6)`  ...[∵ (cos θ + i sin θ)ⁿ = (cos n θ + i sin n θ)]

= `1024[cos(4pi + pi/6) + "i"sin(4pi + pi/6)]`

= `1024(cos  pi/6 - "i"sin  pi/6)`

= `1024[sqrt(3)/2 - 1/2"i"]`

= `512sqrt(3) - 512"i"`