To determine the correct complex number equivalent to the given expression, [tex]\(\frac{1}{3}(6 + 3i) - \frac{2}{3}(6 - 12i)\)[/tex], we'll break it down step-by-step.
1. Calculate [tex]\(\frac{1}{3}(6 + 3i)\)[/tex]:
[tex]\[
\frac{1}{3}(6 + 3i) = \frac{1}{3} \cdot 6 + \frac{1}{3} \cdot 3i = 2 + i
\][/tex]
2. Calculate [tex]\(\frac{2}{3}(6 - 12i)\)[/tex]:
[tex]\[
\frac{2}{3}(6 - 12i) = \frac{2}{3} \cdot 6 + \frac{2}{3} \cdot (-12i) = 4 - 8i
\][/tex]
3. Subtract the two results:
[tex]\[
(2 + i) - (4 - 8i)
\][/tex]
Now, distribute the subtraction:
[tex]\[
2 + i - 4 + 8i = (2 - 4) + (i + 8i) = -2 + 9i
\][/tex]
Therefore, the complex number equivalent to the given expression is [tex]\(-2 + 9i\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{-2 + 9i} \][/tex]
And this matches with:
[tex]\[ \boxed{D} \][/tex]
