Question

Calcule a potência do complexo (√2 + √2 i)3 =

Answer 1

Explicação passo-a-passo:

$\sf (\sqrt{2}+\sqrt{2}\cdot i)^3=[\sqrt{2}\cdot(1+i)]^3$

$\sf (\sqrt{2}+\sqrt{2}\cdot i)^3=(\sqrt{2})^3\cdot(1+i)^3$

Temos que:

• $\sf (1+i)^2=1^2+2\cdot1\cdot i+i^2$

$\sf (1+i)^2=1+2i-1$

$\sf (1+i)^2=2i$

• $\sf (1+i)^3=(1+i)^2\cdot(1+i)$

$\sf (1+i)^3=2i\cdot(1+i)$

$\sf (1+i)^3=2i+2i^2$

$\sf (1+i)^3=2i+2\cdot(-1)$

$\sf (1+i)^3=2i-2$

Logo:

$\sf (\sqrt{2}+\sqrt{2}\cdot i)^3=(\sqrt{2})^2\cdot(2i-2)$

$\sf (\sqrt{2}+\sqrt{2}\cdot i)^3=2\sqrt{2}\cdot(2i-2)$

$\sf (\sqrt{2}+\sqrt{2}\cdot i)^3=-4\sqrt{2}+4\sqrt{2}\cdot i$