Answer
The statement ${{x}^{2}}+{{y}^{2}}=\left( x+yi \right)\left( x-yi \right)$ is true.
Work Step by Step
Consider the expression, $\left( x+yi \right)\left( x-yi \right)$
Use the distributive law $a\left( b+c \right)=ab+ac$.
$\begin{align}
& \left( x+yi \right)\left( x-yi \right)={{x}^{2}}+x\left( -yi \right)+yi\cdot x+yi\left( -yi \right) \\
& ={{x}^{2}}-xyi+xyi-{{y}^{2}}{{i}^{2}} \\
& ={{x}^{2}}-{{y}^{2}}{{i}^{2}}
\end{align}$
Replace the value ${{i}^{2}}=-1$.
$\begin{align}
& \left( x+yi \right)\left( x-yi \right)={{x}^{2}}-{{y}^{2}}\left( -1 \right) \\
& ={{x}^{2}}+{{y}^{2}}
\end{align}$
Therefore, the statement ${{x}^{2}}+{{y}^{2}}=\left( x+yi \right)\left( x-yi \right)$ is true.
