Answer
The statement, “$\left( 3+7i \right)\left( 3-7i \right)$ is an imaginary number” is false. The true statement is, $\left( 3+7i \right)\left( 3-7i \right)$ is a real number.
Work Step by Step
Consider the expression, $\left( 3+7i \right)\left( 3-7i \right)$
Use the FOIL method.
$\begin{align}
& \left( 3+7i \right)\left( 3-7i \right)={{3}^{2}}+3\left( -7i \right)+7i\cdot 3+7i\left( -7i \right) \\
& =9-21i+21i-49{{i}^{2}} \\
& =9-49{{i}^{2}}
\end{align}$
Replace the value ${{i}^{2}}=-1$.
$\left( 3+7i \right)\left( 3-7i \right)=9-49\left( -1 \right)$
Simplify the real terms.
$\begin{align}
& \left( 3+7i \right)\left( 3-7i \right)=9+49 \\
& =58
\end{align}$
The expression $\left( 3+7i \right)\left( 3-7i \right)$ is equal to $58$, which is a real number.
Therefore, the statement, “$\left( 3+7i \right)\left( 3-7i \right)$ is an imaginary number” is false.
