precalculus

Let z = (sqrt(6) - sqrt(2)) + (sqrt(6) + sqrt(2))*i.  Compute z^(24).

 

Convert to polar notation 

 

z = r.exp(i.θ)   where r = √(√6 - √2)² + (√6 + √2)² ) = 4  and  θ = tan^-1( (√6 + √2)/ (√6 - √2) ) = 75°

 

z²⁴ = r²⁴.exp(i.24.θ) = 4²⁴.exp(1800°) = 4²⁴.(cos(1800°) + i.sin(1800°)) = 4²⁴.(cos(0°) + i.sin(0°)) = 4²⁴