Sure, let's work through the given expression step-by-step.
The expression we need to evaluate is:
[tex]\[ 3i(4 + 5i) - 2(3 - 2i) \][/tex]
1. Distribute the complex numbers in both terms:
- First, let's distribute [tex]\(3i\)[/tex] over [tex]\((4 + 5i)\)[/tex]:
[tex]\[
3i(4 + 5i) = 3i \cdot 4 + 3i \cdot 5i
\][/tex]
- [tex]\(3i \cdot 4 = 12i\)[/tex]
- [tex]\(3i \cdot 5i = 15i^2\)[/tex]
Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
15i^2 = 15(-1) = -15
\][/tex]
So, [tex]\(3i(4 + 5i) = 12i - 15\)[/tex].
- Next, let's distribute [tex]\(-2\)[/tex] over [tex]\((3 - 2i)\)[/tex]:
[tex]\[
-2(3 - 2i) = -2 \cdot 3 - (-2) \cdot 2i
\][/tex]
- [tex]\(-2 \cdot 3 = -6\)[/tex]
- [tex]\(-2 \cdot -2i = 4i\)[/tex]
So, [tex]\(-2(3 - 2i) = -6 + 4i\)[/tex].
2. Combine the results:
Now we combine the two results we obtained:
[tex]\[
(12i - 15) - (-6 + 4i)
\][/tex]
Distribute the negative sign through the second term:
[tex]\[
12i - 15 + 6 - 4i
\][/tex]
3. Combine like terms:
- Combine the real parts:
[tex]\[
-15 + 6 = -9
\][/tex]
- Combine the imaginary parts:
[tex]\[
12i - 4i = 8i
\][/tex]
4. Form the final complex number:
[tex]\[
-9 + 8i
\][/tex]
Thus, the result of the operation [tex]\( 3i(4 + 5i) - 2(3 - 2i) \)[/tex] is:
[tex]\[
-9 + 8i
\][/tex]
