To find the value of the product [tex]\(A \cdot B\)[/tex] where [tex]\(A = 6i\)[/tex] and [tex]\(B = 5 + 7i\)[/tex], we will use the distributive property of multiplication over addition for complex numbers. Here is the step-by-step process:
1. Write down the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A = 6i
\][/tex]
[tex]\[
B = 5 + 7i
\][/tex]
2. Apply the distributive property [tex]\((6i) \cdot (5 + 7i)\)[/tex]:
[tex]\[
(6i) \cdot (5 + 7i) = 6i \cdot 5 + 6i \cdot 7i
\][/tex]
3. Calculate each term separately:
[tex]\[
6i \cdot 5 = 30i
\][/tex]
[tex]\[
6i \cdot 7i = 42i^2
\][/tex]
4. Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
42i^2 = 42 \cdot (-1) = -42
\][/tex]
5. Combine the results:
[tex]\[
30i + (-42) = -42 + 30i
\][/tex]
Hence, the value of [tex]\(A \cdot B\)[/tex] is:
[tex]\[
\boxed{-42 + 30i}
\][/tex]
