Write the answer in standard notation.

[tex]\[
(3 \sqrt{3} - 3i)(-4i) = \square
\][/tex]

(Type an exact answer, using radicals and [tex]\( i \)[/tex])



Answer :

To solve the expression [tex]\((3 \sqrt{3} - 3i)(-4i)\)[/tex], we should follow these steps:

1. Distribute [tex]\(-4i\)[/tex] through the binomial [tex]\(3 \sqrt{3} - 3i\)[/tex]:
[tex]\[ (3 \sqrt{3} - 3i)(-4i) = 3 \sqrt{3} \cdot (-4i) + (-3i) \cdot (-4i) \][/tex]

2. Calculate each individual product:

[tex]\[ 3 \sqrt{3} \cdot (-4i) = -12i \sqrt{3} \][/tex]

[tex]\[ (-3i) \cdot (-4i) = 12i^2 \][/tex]

Recall that [tex]\(i^2 = -1\)[/tex]:

[tex]\[ 12i^2 = 12(-1) = -12 \][/tex]

3. Combine the results:

[tex]\[ -12i \sqrt{3} + (-12) = -12 - 12i \sqrt{3} \][/tex]

So, the expression [tex]\((3 \sqrt{3} - 3i)(-4i)\)[/tex] simplifies to:
[tex]\[ -12 - 12i \sqrt{3} \][/tex]
Therefore, the final answer is:
[tex]\[ -12 - 12i \sqrt{3} \][/tex]