evaluate and simplify

Question

⟨-5i²_{-3÷ 6 - 4i¹³}⟩¹²⟩^-2

Answer 1

evaluate and simplify:
( (-5i^2-3)/(6-4i^13) )^12)^(-2) = ?

13/4 = 3 + 1/4
i^13 = i

(-5i^2-3)/(6-4i^13)
= (5-3)/(6-4i)
= 2/(6-4i)
= 1/(3-2i)

( (-5i^2-3)/(6-4i^13) )^12)^(-2)
= ( 1/(3-2i) )^(-24)
= (3-2i)^(24)

|3-2i| = √(3^2+(-2)^2) = √(13)
θ = arctan(-2/3) ≈ -0.58800260354757 ≈ -0.187167041811 pi
(3-2i) = √(13) e^(i θ)
(3-2i)^(24) = (√(13))^(24) e^(i(24θ))
= (13)^(12) e^(i(24 θ))
24 θ ≈ -4.492009003464 pi + 6 pi
≈ 1.507990996536 pi ≈ 3pi/2

13^12 e^(i(24 θ)) ≈ 13^12 (cos(3pi/2) + i sin(3pi/2) )
≈ 13^12 (0 + i (-1) )

≈ –23,298,085,122,481 i <== answer