Let's solve these problems step-by-step using properties of complex numbers.
### Problem 1: [tex]\((4 - 5i)(4 + 5i)\)[/tex]
1. When multiplying two complex conjugates, [tex]\((a + bi)(a - bi) = a^2 + b^2\)[/tex].
2. Here, [tex]\(a = 4\)[/tex] and [tex]\(b = 5\)[/tex].
3. Using the formula:
[tex]\[
(4 - 5i)(4 + 5i) = 4^2 + 5^2
\][/tex]
4. Calculate the squares:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
5^2 = 25
\][/tex]
5. Add the results:
[tex]\[
4^2 + 5^2 = 16 + 25 = 41
\][/tex]
Thus, [tex]\((4 - 5i)(4 + 5i) = 41\)[/tex]. Since we are dealing with real numbers here, we can write the result as:
[tex]\[
(4 - 5i)(4 + 5i) = 41
\][/tex]
### Problem 2: [tex]\((-3 + 8i)(-3 - 8i)\)[/tex]
1. Similarly, we use the property of complex conjugates, [tex]\((a + bi)(a - bi) = a^2 + b^2\)[/tex].
2. Here, [tex]\(a = -3\)[/tex] and [tex]\(b = 8\)[/tex].
3. Using the formula:
[tex]\[
(-3 + 8i)(-3 - 8i) = (-3)^2 + 8^2
\][/tex]
4. Calculate the squares:
[tex]\[
(-3)^2 = 9
\][/tex]
[tex]\[
8^2 = 64
\][/tex]
5. Add the results:
[tex]\[
(-3)^2 + 8^2 = 9 + 64 = 73
\][/tex]
Thus, [tex]\((-3 + 8i)(-3 - 8i) = 73\)[/tex]. Since we are dealing with real numbers here, we can write the result as:
[tex]\[
(-3 + 8i)(-3 - 8i) = 73
\][/tex]
### Summary
[tex]\[
(4 - 5i)(4 + 5i) = 41
\][/tex]
[tex]\[
(-3 + 8i)(-3 - 8i) = 73
\][/tex]
