Answer
The standard form of the expression ${{\left( 3i-4 \right)}^{2}}$ is $-7-24i$.
Work Step by Step
Consider the expression,
${{\left( 3i-4 \right)}^{2}}$
Apply the square of the difference of the square property on the expression ${{\left( 3i-4 \right)}^{2}}$
Therefore,
$\begin{align}
& {{\left( 3i-4 \right)}^{2}}={{\left( 3i \right)}^{2}}-2\left( 3i \right)+{{\left( 4 \right)}^{2}} \\
& =9{{i}^{2}}-6i+16
\end{align}$
As ${{i}^{2}}=-1$
Therefore,
$\begin{align}
& 9{{i}^{2}}-24i+16=9\left( -1 \right)-24i+16 \\
& =-9-24i+16
\end{align}$
Combine the real part and imaginary part separately and either add or subtract as required.
$\begin{align}
& -9-24i+16=\left( -9+16 \right)-24i \\
& =-7-24i
\end{align}$
Hence, the standard form of the expression ${{\left( 3i-4 \right)}^{2}}$ is$-7-24i$.
