To solve for [tex]\((3 + 2i)^3\)[/tex], we will follow these steps:
1. Start by expressing the given complex number [tex]\(3 + 2i\)[/tex].
2. Apply the formula for raising a complex number to a power: [tex]\((a + bi)^3\)[/tex], where [tex]\(a = 3\)[/tex] and [tex]\(b = 2\)[/tex].
3. Raise the complex number [tex]\((3 + 2i)\)[/tex] to the power of three and compute the result in the form [tex]\(a + bi\)[/tex].
When you carry out the steps, the result is:
[tex]\[
(3 + 2i)^3 = -9 + 46i
\][/tex]
So, the real part of [tex]\((3 + 2i)^3\)[/tex] is [tex]\(-9\)[/tex] and the imaginary part is [tex]\(46\)[/tex].
Therefore, the correct answer is:
(b) [tex]\(46i - 9\)[/tex]
