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If $z=3+5 i$, then $z^3+\bar{z}+198$ is equal to
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2758 Upvotes
Verified Answer
The correct answer is:
$3+5 i$
We have,
$$
\begin{aligned}
z & =3+5 i \\
\therefore \quad \bar{z} & =3-5 i \\
z^3 & =z^2 \cdot z=(3+5 i)^2(3+5 i) \\
& =(9-25+30 i)(3+5 i) \\
& =(-16+30 i)(3+5 i) \\
& =-48-150+10 i \\
& =-198+10 i
\end{aligned}
$$
Now,
$$
\begin{aligned}
z^3+\bar{z}+198 & =-198+10 \dot{i}+3-5 i+198 \\
& =3+5 i
\end{aligned}
$$
$$
\begin{aligned}
z & =3+5 i \\
\therefore \quad \bar{z} & =3-5 i \\
z^3 & =z^2 \cdot z=(3+5 i)^2(3+5 i) \\
& =(9-25+30 i)(3+5 i) \\
& =(-16+30 i)(3+5 i) \\
& =-48-150+10 i \\
& =-198+10 i
\end{aligned}
$$
Now,
$$
\begin{aligned}
z^3+\bar{z}+198 & =-198+10 \dot{i}+3-5 i+198 \\
& =3+5 i
\end{aligned}
$$
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