# Chapter 2 - Section 2.1 - Complex Numbers - Exercise Set - Page 315: 78

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.

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