Question

Números complexos
calcule:
(1/15 + 1/15i) elevado a 8

Answer 1

$\left(\dfrac{1}{15}+\dfrac{1}{15}i \right )^{8}\\ \\ =\left[\,\dfrac{1}{15}\left(1+i \right )\, \right ]^{8}\\ \\ =\left(\dfrac{1}{15} \right )^{8}\cdot \left(1+i \right )^{8}\\ \\ =\dfrac{1}{15^{8}}\cdot \left(1+i \right )^{2\, \cdot\, 4}\\ \\ =\dfrac{1}{15^{8}}\cdot \left[\,\left(1+i \right )^{2}\, \right ]^{4}\\ \\ =\dfrac{1}{15^{8}}\cdot \left[\,1^{2}+2i+i^{2}\, \right ]^{4}\\ \\ =\dfrac{1}{15^{8}}\cdot \left[\,1+2i+\left(-1 \right )\, \right ]^{4}\\ \\ =\dfrac{1}{15^{8}}\cdot \left[\,2i\, \right ]^{4}\\ \\ =\dfrac{1}{15^{8}}\cdot 2^{4}i^{4}\\ \\ =\dfrac{2^{4}}{15^{8}}\cdot 1\\ \\ =\dfrac{2^{4}}{15^{8}}\\ \\ \\ \boxed{\left(\dfrac{1}{15}+\dfrac{1}{15}i \right )^{8}=\dfrac{2^{4}}{15^{8}}}$