To multiply the complex numbers [tex]\( (4 - 3i) \)[/tex] and [tex]\( (2 + 6i) \)[/tex], follow these steps:
1. Use the distributive property (also known as the FOIL method for binomials) to expand the product:
[tex]\[
(4 - 3i) \cdot (2 + 6i) = 4 \cdot 2 + 4 \cdot 6i - 3i \cdot 2 - 3i \cdot 6i
\][/tex]
2. Calculate each term individually:
[tex]\[
4 \cdot 2 = 8
\][/tex]
[tex]\[
4 \cdot 6i = 24i
\][/tex]
[tex]\[
-3i \cdot 2 = -6i
\][/tex]
[tex]\[
-3i \cdot 6i = -18i^2
\][/tex]
3. Recall that [tex]\(i^2 = -1\)[/tex], so convert [tex]\( -18i^2 \)[/tex]:
[tex]\[
-18i^2 = -18(-1) = 18
\][/tex]
4. Combine all terms together:
[tex]\[
8 + 24i - 6i + 18
\][/tex]
5. Simplify the expression by combining like terms:
[tex]\[
(8 + 18) + (24i - 6i)
\][/tex]
[tex]\[
26 + 18i
\][/tex]
Therefore, the product of the complex numbers [tex]\( (4 - 3i) \cdot (2 + 6i) \)[/tex] is [tex]\( 26 + 18i \)[/tex].
