Question

select the correct answer.
which complex number is equivalent to this expression?
\\(\frac{1}{3}(6 + 3\mathrm{i}) - \frac{2}{3}(6 - 12\mathrm{i})\\)

a. \\(6 + 9\mathrm{i}\\)
b. \\(6 - 7\mathrm{i}\\)
c. \\(-2 - 7\mathrm{i}\\)
d. \\(-2 + 9\mathrm{i}\\)

Explanation:

Step1: Distribute the fractions

First, distribute \(\frac{1}{3}\) to \(6 + 3i\) and \(\frac{2}{3}\) to \(6 - 12i\).
For \(\frac{1}{3}(6 + 3i)\), we have \(\frac{1}{3} \times 6 + \frac{1}{3} \times 3i = 2 + i\).
For \(\frac{2}{3}(6 - 12i)\), we have \(\frac{2}{3} \times 6 - \frac{2}{3} \times 12i = 4 - 8i\).

Step2: Subtract the two complex numbers

Now, we need to compute \((2 + i) - (4 - 8i)\).
Subtract the real parts and the imaginary parts separately:
Real part: \(2 - 4 = -2\)
Imaginary part: \(i - (-8i) = i + 8i = 9i\)

So, combining the real and imaginary parts, we get \(-2 + 9i\).

Answer:

D. -2 + 9i