Answer
The standard form of the expression ${{\left( -2+\sqrt{-100} \right)}^{2}}$ is $-96-40i$.
Work Step by Step
Consider the expression,
${{\left( -2+\sqrt{-100} \right)}^{2}}$
Rewrite the expression ${{\left( -2+\sqrt{-100} \right)}^{2}}$ as ${{\left( -2+\sqrt{100}\sqrt{-1} \right)}^{2}}$
As $\sqrt{-1}=i$
Therefore,
${{\left( -2+\sqrt{100}\sqrt{-1} \right)}^{2}}={{\left( -2+10i \right)}^{2}}$
Apply the square of the difference on the above expression.
$\begin{align}
& {{\left( -2+10i \right)}^{2}}={{2}^{2}}-2\left( 2 \right)\left( 10i \right)+{{\left( 10i \right)}^{2}} \\
& =4-40i+100{{i}^{2}}
\end{align}$
As ${{i}^{2}}=-1$
Therefore,
$\begin{align}
& 4-40i+100{{i}^{2}}=4-40i+100\left( -1 \right) \\
& =4-40i-100
\end{align}$
So,
${{\left( -2+10i \right)}^{2}}=4-40i-100$
To subtract two complex numbers, combine the real numbers together and the terms containing $i$.
$\begin{align}
& 4-40i-100=\left( 4-100 \right)-40i \\
& =-96-40i
\end{align}$
Hence, the standard form of the expression ${{\left( -2+\sqrt{-100} \right)}^{2}}$ is $-96-40i$.
