To find the complex number equivalent to the expression given, we start by breaking it down into two parts and simplifying each one separately.
The expression is:
[tex]\[
\frac{1}{3}(6+3 i) - \frac{2}{3}(6-12 i)
\][/tex]
1. Simplify the first term:
[tex]\[
\frac{1}{3}(6+3i)
\][/tex]
We can distribute [tex]\(\frac{1}{3}\)[/tex] across the terms inside the parentheses:
[tex]\[
\frac{1}{3} \cdot 6 + \frac{1}{3} \cdot 3i = 2 + i
\][/tex]
So, the first term simplifies to [tex]\(2 + i\)[/tex].
2. Simplify the second term:
[tex]\[
\frac{2}{3}(6-12i)
\][/tex]
We distribute [tex]\(\frac{2}{3}\)[/tex] across the terms inside the parentheses:
[tex]\[
\frac{2}{3} \cdot 6 - \frac{2}{3} \cdot 12i = 4 - 8i
\][/tex]
So, the second term simplifies to [tex]\(4 - 8i\)[/tex].
3. Subtract the second term from the first term:
Combine [tex]\(2 + i\)[/tex] and [tex]\(4 - 8i\)[/tex]:
[tex]\[
(2 + i) - (4 - 8i) = 2 + i - 4 + 8i
\][/tex]
Combine like terms:
[tex]\[
= (2 - 4) + (i + 8i) = -2 + 9i
\][/tex]
Therefore, the complex number equivalent to the given expression is [tex]\(-2 + 9i\)[/tex].
4. The correct answer is:
[tex]\[
\boxed{-2 + 9i}
\][/tex]
So, the correct option is:
D. [tex]\( -2 + 9i \)[/tex]
