Question

Sendo z = 5i + 3i^2 - 2i^3 + 4i^27 e w = 2i^12 - 3i^15 , calcule Im(z).w + Im(w).z

Answer 1

$Z=5i+3i^2-2i^3+4i^{27}\\\\Z=5i+3\cdot(-1)-2\cdot(-i)+4\cdot(-i)\\\\Z=5i-3+2i-4i\\\\\boxed{Z=-3+3i}$

$W=2i^{12}-3i^{15}\\\\W=2\cdot1-3\cdot(-i)\\\\\boxed{W=2+3i}$

$\mbox{Im}(Z)\cdot W+\mbox{Im}(W)\cdot Z$

Sabendo que Im corresponde apenas à parte imaginária do número complexo:

$3i\cdot(2+3i)+3i\cdot(-3+3i)\\\\6i+9i^2-9i+9i^2\\\\18i^2-3i\\\\\boxed{-18-3i}$

Answer 2

Boa tarde Lauro 

potencias de i

i^(4k+0) = 1
i^(4k+1) = i
i^(4k+2) = -1
i^(4k+3) = -i


z = 5i + 3i² - 2i³ + 4i²⁷ = 5i - 3 + 2i - 4i = -3 + 3i 

w = 2i¹² - 3i¹⁵ = 2 + 3i 

Im(z) = 3i
Im(w) = 3i 

Im(z)*w + Im(w)*z = 3i*(2 + 3i - 3 + 3i) = 3i*(-1 + 6i) = -18 - 3i