Question

Resolver as divisões com números complexos: a) ( 2+3i) / ( 5 -2i) b) ( 3 - 8i) / ( 19 + 2i) c) ( 4 + 6i) / ( 21 - 7i) d) ( 5 - 2i) / ( 8 +3i) e)( 6 + 6i) /( 7 +6i) f) ( 9 - 7i) / ( 8 -2i)

Answer 1

Explicação passo-a-passo:

a)

$\sf \dfrac{2+3i}{5-2i}$

$\sf =\dfrac{2+3i}{5-2i}\cdot\dfrac{5+2i}{5+2i}$

$\sf =\dfrac{10+4i+15i+6i^2}{5^2-(2i)^2}$

$\sf =\dfrac{10+19i+6\cdot(-1)}{25-4i^2}$

$\sf =\dfrac{10+19i-6}{25-4\cdot(-1)}$

$\sf =\dfrac{4+19i}{25+4}$

$\sf =\dfrac{4+19i}{29}$

$\sf =\red{\dfrac{4}{29}+\dfrac{19i}{29}}$

b) $\sf \dfrac{3-8i}{19+2i}$

$\sf =\dfrac{3-8i}{19+2i}\cdot\dfrac{19-2i}{19-2i}$

$\sf =\dfrac{57-6i-152+16i^2}{19^2-(2i)^2}$

$\sf =\dfrac{57-158i+16\cdot(-1)}{361-4i^2}$

$\sf =\dfrac{57-158i-16}{361-4\cdot(-1)}$

$\sf =\dfrac{41-158i}{361+4}$

$\sf =\dfrac{41-158i}{365}$

$\sf =\red{\dfrac{41}{365}-\dfrac{158i}{365}}$

c) $\sf \dfrac{4+6i}{21-7i}$

$\sf =\dfrac{4+6i}{21-7i}\cdot\dfrac{21+7i}{21+7i}$

$\sf =\dfrac{84+28i+126i+42i^2}{21^2-(7i)^2}$

$\sf =\dfrac{84+154i+42\cdot(-1)}{441-49i^2}$

$\sf =\dfrac{84+154i-42}{441-49\cdot(-1)}$

$\sf =\dfrac{42+154i}{441+49}$

$\sf =\dfrac{42+154i}{490}$

$\sf =\dfrac{42}{490}+\dfrac{154i}{490}$

$\sf =\red{\dfrac{3}{35}+\dfrac{11i}{35}}$

d) $\sf \dfrac{5-2i}{8+3i}$

$\sf =\dfrac{5-2i}{8+3i}\cdot\dfrac{8-3i}{8-3i}$

$\sf =\dfrac{40-15i-16i+6i^2}{8^2-(3i)^2}$

$\sf =\dfrac{40-31i+6\cdot(-1)}{64-9i^2}$

$\sf =\dfrac{40-31i-6}{64-9\cdot(-1)}$

$\sf =\dfrac{34-31i}{64+9}$

$\sf =\dfrac{34-31i}{73}$

$\sf =\red{\dfrac{34}{73}-\dfrac{31i}{73}}$

e) $\sf \dfrac{6+6i}{7+6i}$

$\sf =\dfrac{6+6i}{7+6i}\cdot\dfrac{7-6i}{7-6i}$

$\sf =\dfrac{42-36i+42i-36i^2}{7^2-(6i)^2}$

$\sf =\dfrac{42+6i-36\cdot(-1)}{49-36i^2}$

$\sf =\dfrac{42+6i+36}{49-36\cdot(-1)}$

$\sf =\dfrac{78+6i}{49+36}$

$\sf =\dfrac{78+6i}{85}$

$\sf =\red{\dfrac{78}{85}+\dfrac{6i}{85}}$

f) $\sf \dfrac{9-7i}{8-2i}$

$\sf =\dfrac{9-7i}{8-2i}\cdot\dfrac{8+2i}{8+2i}$

$\sf =\dfrac{72+18i-56i-14i^2}{8^2-(2i)^2}$

$\sf =\dfrac{72-38i-14\cdot(-1)}{64-4i^2}$

$\sf =\dfrac{72-38i+14}{64-4\cdot(-1)}$

$\sf =\dfrac{86-38i}{64+4}$

$\sf =\dfrac{86-38i}{68}$

$\sf =\dfrac{86}{68}-\dfrac{38i}{68}$

$\sf =\red{\dfrac{43}{34}-\dfrac{19i}{34}}$