Question

Calcule:
$( \sqrt{3} - i) {}^{10}$
Ajudem pfvr.

Answer 1

Vamos primeiro calcular $(\sqrt{3}-i)^3$

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

$(\sqrt{3}-i)^3=\sqrt{3^3}-3\cdot3\cdot i+3\sqrt{3}\cdot(-1)-(-i)$

$(\sqrt{3}-i)^3=3\sqrt{3}-9i-3\sqrt{3}+i$

$(\sqrt{3}-i)^{3}=-8i$

Veja que:

$(\sqrt{3}-i)^{10}=(\sqrt{3}-i)^{9}\cdot(\sqrt{3}-i)^{1}=[(\sqrt{3}-i)^3]^{3}\cdot(\sqrt{3}-i)^{1}$

Como $(\sqrt{3}-i)^{3}=-8i$, temos que:

$(\sqrt{3}-i)^{10}=(-8i)^3\cdot(\sqrt{3}-i)^{1}$

Observe que $(-8i)^3=(-1)^3\cdot8^3\cdot i^3=(-1)\cdot512\cdot(-i)=512i$

Logo:

$(-8i)^3\cdot(\sqrt{3}-i)^{1}=512i\cdot(\sqrt{3}-i)=512\sqrt{3}\cdot i-512i^2=512\sqrt{3}\cdot i+512$

Portanto, $(\sqrt{3}-i)^{10}=512+512\sqrt{3}\cdot i$