Show that (-1+ 3 i)^3 is a real number. - Vidyadip

Question

\text { Show that }(-1+\sqrt{ } 3 i)^3 \text { is a real number. }

✓

Answer

\begin{aligned}
& (-1+\sqrt{ } 3 i)^3 \\
& =(-1)^3+3(-1)^2(\sqrt{ } 3 i)+3(-1)(\sqrt{ } 3 i)^2+(\sqrt{ } 3 i)^3\left[\because(a+b)^3=a^3+3 a^2 b+3 a b^2+b^3\right] \\
& =-1+3 \sqrt{ } 3 i-3\left(3 i^2\right)+3 \sqrt{ } 3 i^3 \\
& =-1+3 \sqrt{ } 3 i-3(-3)-3 \sqrt{ } 3 i\left[\because i^2=-1, i^3=-1\right] \\
& =-1+9 \\
& =8 \text {, which is a real number. }
\end{aligned}

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