To find the product of the complex numbers [tex]\( (9 + 7i) \)[/tex] and [tex]\( (6 + 3i) \)[/tex], we follow these steps:
1. Distribute each term in the first complex number by each term in the second complex number:
[tex]\[
(9 + 7i)(6 + 3i) = 9 \cdot 6 + 9 \cdot 3i + 7i \cdot 6 + 7i \cdot 3i
\][/tex]
2. Multiply each term:
[tex]\[
9 \cdot 6 = 54
\][/tex]
[tex]\[
9 \cdot 3i = 27i
\][/tex]
[tex]\[
7i \cdot 6 = 42i
\][/tex]
[tex]\[
7i \cdot 3i = 21i^2
\][/tex]
3. Combine the terms:
[tex]\[
54 + 27i + 42i + 21i^2
\][/tex]
4. Recall that [tex]\( i^2 = -1 \)[/tex] for imaginary numbers:
[tex]\[
21i^2 = 21(-1) = -21
\][/tex]
5. Substitute [tex]\(-21\)[/tex] for [tex]\( 21i^2 \)[/tex] and combine the real and imaginary parts:
[tex]\[
54 + 27i + 42i - 21
\][/tex]
6. Combine the real parts:
[tex]\[
54 - 21 = 33
\][/tex]
7. Combine the imaginary parts:
[tex]\[
27i + 42i = 69i
\][/tex]
So, combining the real and imaginary parts together, the product of the complex numbers is [tex]\( 33 + 69i \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{33 + 69i}
\][/tex]
So, the correct choice is:
B. [tex]\(33 + 69i\)[/tex]
