Write the complex conjugate of the following complex number and then multiply the number by the complex conjugate. Write the result in standard form.
$2 + \sqrt { - 27 }$
A) $2 - 27 i ; 25$
B) $2 + 3 \sqrt { 3 } i ; 31$
C) $- 2 - 3 \sqrt { 3 } i ; 31$
D) $- 2 - 3 \sqrt { 3 } i ; \quad 25$
E) $2 - 3 \sqrt { 3 } i ; 31$

Question

Quizplus · Accepted Answer

The complex conjugate of $2 + \sqrt{-27}$ is $2 - \sqrt{-27}$ or $2 - 3\sqrt{3}i$, since $\sqrt{-27} = 3\sqrt{3}i$. Multiplying the number by its complex conjugate gives $ (2 + 3\sqrt{3}i)(2 - 3\sqrt{3}i) = 4 - 27 = -23$. However, there seems to be a mistake in my calculation or interpretation of the options, as none directly match the expected process. Correctly, multiplying the complex number by its conjugate should yield a positive result due to the nature of $i^2 = -1$. The correct process is $ (2 + 3\sqrt{3}i)(2 - 3\sqrt{3}i) = 4 + 27 = 31$, aligning with option E.

Write the Complex Conjugate of the Following Complex Number and Then