Question

A fração i³ - i² + i¹7 - i³5 sobre i¹6 - i¹³ + i³0 corresponde a qual numero complexo?

Resposta: -1 + i

Answer 1

i° = 1
i¹ = i
i² = -1
i³ = -i

y así se repite $i^{4n}=1\;,\; i^{4n+1}=i\;,\; i^{4n+2}=-1\;,\; i^{4n+3}=-i$

entonces

          $\dfrac{i^3-i^2+i^{17}-i^{35}}{i^{16}-i^{13}+i^{10}}=\dfrac{-i-(-1)+i^{4(4)+1}-i^{4(8)+3}}{i^{4(4)}-i^{4(3)+1}+i^{4(2)+2}}\\ \\ \\ \dfrac{i^3-i^2+i^{17}-i^{35}}{i^{16}-i^{13}+i^{10}}=\dfrac{-i+1+i-(-i)}{1-i-1}\\ \\ \\ \dfrac{i^3-i^2+i^{17}-i^{35}}{i^{16}-i^{13}+i^{10}}=\dfrac{1+i}{-i}\\ \\ \\ \boxed{\dfrac{i^3-i^2+i^{17}-i^{35}}{i^{16}-i^{13}+i^{10}}=-1+i} \\ \\.$

Answer 2

Oi Marcos

i^3 = -i 
i^2 = -1
i^17 = i
i^35 = -i 

N = -i  + 1 + i + i  = 1 + i

i^16 = 1
i^13 = i
i^30 = -1

D = 1 - i - 1 = -i

E = N/D = (1 + i)/-i  = (i + i²)/-i² = (i - 1)/1 = -1 + i