To simplify the expression [tex]\((6 + i)(8 - 3i)\)[/tex], we can use the distributive property to expand the product. Let's go through the steps in detail:
1. Distribution:
[tex]\[
(6 + i)(8 - 3i) = 6 \cdot 8 + 6 \cdot (-3i) + i \cdot 8 + i \cdot (-3i)
\][/tex]
2. Multiply each term:
- [tex]\(6 \cdot 8 = 48\)[/tex]
- [tex]\(6 \cdot (-3i) = -18i\)[/tex]
- [tex]\(i \cdot 8 = 8i\)[/tex]
- [tex]\(i \cdot (-3i) = -3i^2\)[/tex]
To find [tex]\(-3i^2\)[/tex], recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
-3i^2 = -3(-1) = 3
\][/tex]
3. Combine the terms:
- Real parts: [tex]\(48 + 3 = 51\)[/tex]
- Imaginary parts: [tex]\(-18i + 8i = -10i\)[/tex]
4. Construct the simplified form:
[tex]\[
(6 + i)(8 - 3i) = 51 - 10i
\][/tex]
Thus, the simplified form of [tex]\((6 + i)(8 - 3i)\)[/tex] is [tex]\(51 - 10i\)[/tex].
Comparing it with the given options:
- [tex]\(51 - 10i\)[/tex]
- [tex]\(45 - 10i\)[/tex]
- [tex]\(48 - 3i^2\)[/tex]
- [tex]\(48 - 10i - 3i^2\)[/tex]
The correct answer is [tex]\(\boxed{51 - 10i}\)[/tex].
