Question

C-calcule........... ​

Answer 1

$\large\boxed{\begin{array}{l}\tt a)~\sf 163=40\cdot4+\!\!\!\!\!\!\underbrace{\boxed{\sf 3}}_{resto\,da\,divis\tilde ao}\\\sf i^{163}=i^{3}=-i\\\tt b)~\sf 8=4\cdot2+0\\\sf 4i^8=4\cdot i^0=4\cdot1=4\\\tt c)~\sf5=2\cdot2+1\\\sf 2i^5=2i^1=2i\\\tt d)~\sf(3+2i)^3=3^3+3\cdot 3^2\cdot2i+3\cdot3\cdot(2i)^2+(2i)^3\\\sf(3+2i)^3=27+54i+9\cdot4i^2+8i^3\\\sf (3+2i)^3=27+54i-36-8i\\\sf ( 3+2i)^3=-9+46i\end{array}}$

$\large\boxed{\begin{array}{l}\tt e)~\sf(4-3i)^3=4^3-3\cdot4^2\cdot3i+3\cdot4\cdot(3i)^2-(3i)^3\\\sf (4-3i)^3=64-3\cdot16\cdot3i+3\cdot4\cdot9i^2-27i^3\\\sf (4-3i)^3=64-144i-108+27i\\\sf (4-3i)^3=-44-117i\end{array}}$

$\Large\boxed{\begin{array}{l}\tt f)~\sf(1-i)^2=1-2i+i^2\\\sf (1-i)^2=1-2i-1=-2i\\\\\tt g)~\sf\bigg(-\dfrac{1}{2}+\dfrac{3}{2}i\bigg)^2=\dfrac{1}{4}-\dfrac{3}{2}i+\dfrac{9}{4}i^2\\\\\sf\bigg(-\dfrac{1}{2}+\dfrac{3}{2}i\bigg)^2=\dfrac{1}{4}-\dfrac{3}{2}i-\dfrac{9}{4}=\dfrac{1-9}{4}-\dfrac{3}{2}i\\\\\sf\bigg(-\dfrac{1}{2}-\dfrac{3}{2}\bigg)^2=-2-\dfrac{3}{2}i\end{array}}$

$\Large\boxed{\begin{array}{l}\tt h)~\sf(-3+i)^3=-(i-3)^3\\\sf (i-3)^3=i^3-3\cdot i^2\cdot3+3\cdot i\cdot 3^2-3^3\\\sf (i-3)^3=-i+9+27i-27\\\sf (i-3)^3=-18+26i\\\sf (-3+i)^3=18-26i\end{array}}$