Answer
$(a+bi)(a-bi)=a^{2}-b^{2}i^{2}=a^{2}-b^{2}(-1)=a^{2}+b^{2}$
Work Step by Step
If the complex number z is represented by $a+bi$ then its conjugate is $a-bi$.
Multiply the complex number and its conjugate:
$(a+bi)(a-bi)$
=$a(a-bi)+bi(a-bi)$
=$a^{2}-abi+abi-b^{2}i^{2}$
=$a^{2}-b^{2}i^{2}$
=$a^{2}-b^{2}(-1)$
=$a^{2}+b^{2}$.
Since both $a$ and $b$ are real numbers, the product of z and its conjugate is a real number.
