Precalculus (6th Edition) Blitzer

The standard form of the expression ${{\left( -2+\sqrt{-100} \right)}^{2}}$ is $-96-40i$.
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 $-96-40i$.