Sure, let's simplify each of the given expressions step-by-step.
### Part (a)
Expression:
[tex]\[
(3 + 7i) + (2 - 4i)
\][/tex]
To add complex numbers, we add the real parts and the imaginary parts separately.
Step-by-Step Solution:
1. Real part: [tex]\(3 + 2 = 5\)[/tex]
2. Imaginary part: [tex]\(7i + (-4i) = 3i\)[/tex]
Simplified Form:
[tex]\[
5 + 3i
\][/tex]
### Part (b)
Expression:
[tex]\[
(7 + i) - (2 - 4i)
\][/tex]
To subtract complex numbers, we subtract the real parts and the imaginary parts separately.
Step-by-Step Solution:
1. Real part: [tex]\(7 - 2 = 5\)[/tex]
2. Imaginary part: [tex]\(i - (-4i) = i + 4i = 5i\)[/tex]
Simplified Form:
[tex]\[
5 + 5i
\][/tex]
### Part (c)
Expression:
[tex]\[
(5 + 2i)(4 - 3i)
\][/tex]
To multiply complex numbers, we use the distributive property (FOIL method) to expand the product, remembering that [tex]\(i^2 = -1\)[/tex].
Step-by-Step Solution:
1. First, we expand:
[tex]\[
(5 + 2i)(4 - 3i) = 5 \cdot 4 + 5 \cdot (-3i) + 2i \cdot 4 + 2i \cdot (-3i)
\][/tex]
2. Perform the individual multiplications:
[tex]\[
= 20 - 15i + 8i - 6i^2
\][/tex]
3. Combine the imaginary terms:
[tex]\[
= 20 - 7i - 6i^2
\][/tex]
4. Replace [tex]\(i^2\)[/tex] with [tex]\(-1\)[/tex]:
[tex]\[
-6i^2 = -6(-1) = 6
\][/tex]
5. Simplify by combining the real parts:
[tex]\[
= 20 + 6 - 7i = 26 - 7i
\][/tex]
Simplified Form:
[tex]\[
26 - 7i
\][/tex]
### Summary of Answers:
a. [tex]\(5 + 3i\)[/tex]
b. [tex]\(5 + 5i\)[/tex]
c. [tex]\(26 - 7i\)[/tex]
