To reduce the expression [tex]\((3 - i)(4 + 5i)(6 + 2i)\)[/tex] into the form [tex]\(a + bi\)[/tex], follow these steps:
1. Multiply [tex]\((3 - i)\)[/tex] and [tex]\((4 + 5i)\)[/tex] first:
Let's denote:
[tex]\[
z_1 = 3 - i
\][/tex]
[tex]\[
z_2 = 4 + 5i
\][/tex]
Calculate [tex]\(z_1 \cdot z_2\)[/tex]:
[tex]\[
(3 - i)(4 + 5i) = 3 \cdot 4 + 3 \cdot 5i - i \cdot 4 - i \cdot 5i
\][/tex]
Simplify step-by-step:
[tex]\[
= 12 + 15i - 4i - 5i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
= 12 + 15i - 4i + 5
\][/tex]
Combine like terms:
[tex]\[
= (12 + 5) + (15i - 4i)
\][/tex]
[tex]\[
= 17 + 11i
\][/tex]
2. Now multiply the result [tex]\((17 + 11i)\)[/tex] with [tex]\((6 + 2i)\)[/tex]:
Let’s denote:
[tex]\[
z_3 = 6 + 2i
\][/tex]
Calculate [tex]\((17 + 11i) \cdot (6 + 2i)\)[/tex]:
[tex]\[
(17 + 11i)(6 + 2i) = 17 \cdot 6 + 17 \cdot 2i + 11i \cdot 6 + 11i \cdot 2i
\][/tex]
Simplify step-by-step:
[tex]\[
= 102 + 34i + 66i + 22i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
= 102 + 34i + 66i + 22(-1)
\][/tex]
[tex]\[
= 102 + 100i - 22
\][/tex]
Combine like terms:
[tex]\[
= (102 - 22) + 100i
\][/tex]
[tex]\[
= 80 + 100i
\][/tex]
So, the result in the form [tex]\(a + bi\)[/tex] is:
[tex]\[
80 + 100i
\][/tex]
Therefore, [tex]\(a = 80\)[/tex] and [tex]\(b = 100\)[/tex].
