If z=2+3i, then z^5+ z^5 is equal to:

Question

If $z=2+3i$, then $z^5+\bar z^5$ is equal to:

Accepted Answer

$z^2=(2+3i)^2=4+12i-9=-5+12i.$
$$z^4=(z^2)^2=(-5+12i)^2=25-120i-144=-119-120i.$$
$$z^5=z\cdot z^4=(2+3i)(-119-120i)=-238-240i-357i-360i^2=122-597i.$$

$\bar z^5$ is the conjugate: $122+597i$.

$$z^5+\bar z^5=2(122)=244.$$

Solutions · 1