Answer :
Sure! Let's work through the steps to multiply and simplify [tex]\((8 - 5i)^2\)[/tex].
1. Expand the expression:
[tex]\[
(8 - 5i)^2 = (8 - 5i)(8 - 5i)
\][/tex]
2. Use the distributive property to expand:
[tex]\[
(8 - 5i)(8 - 5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i)
\][/tex]
3. Perform the multiplications:
[tex]\[
8 \cdot 8 = 64
\][/tex]
[tex]\[
8 \cdot (-5i) = -40i
\][/tex]
[tex]\[
(-5i) \cdot 8 = -40i
\][/tex]
[tex]\[
(-5i) \cdot (-5i) = 25i^2
\][/tex]
4. Simplify the term involving [tex]\(i^2\)[/tex]:
Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Combine all terms:
[tex]\[
64 - 40i - 40i - 25
\][/tex]
6. Combine like terms:
[tex]\[
64 - 25 = 39 \quad \text{(real part)}
\][/tex]
[tex]\[
-40i - 40i = -80i \quad \text{(imaginary part)}
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
1. Expand the expression:
[tex]\[
(8 - 5i)^2 = (8 - 5i)(8 - 5i)
\][/tex]
2. Use the distributive property to expand:
[tex]\[
(8 - 5i)(8 - 5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i)
\][/tex]
3. Perform the multiplications:
[tex]\[
8 \cdot 8 = 64
\][/tex]
[tex]\[
8 \cdot (-5i) = -40i
\][/tex]
[tex]\[
(-5i) \cdot 8 = -40i
\][/tex]
[tex]\[
(-5i) \cdot (-5i) = 25i^2
\][/tex]
4. Simplify the term involving [tex]\(i^2\)[/tex]:
Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Combine all terms:
[tex]\[
64 - 40i - 40i - 25
\][/tex]
6. Combine like terms:
[tex]\[
64 - 25 = 39 \quad \text{(real part)}
\][/tex]
[tex]\[
-40i - 40i = -80i \quad \text{(imaginary part)}
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
