Answer :
Sure! Let's multiply and simplify [tex]\((8 - 5i)^2\)[/tex] step by step.
1. Start by using the formula for squaring a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
In our case, [tex]\( a = 8 \)[/tex] and [tex]\( b = 5i \)[/tex].
2. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
3. Calculate each term separately:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
2 \cdot 8 \cdot 5i = 80i
\][/tex]
[tex]\[
(5i)^2 = 25i^2
\][/tex]
4. Recall that [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
25i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Now, substitute these back into the equation:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
6. Combine the real parts and the imaginary parts:
[tex]\[
64 - 25 = 39
\][/tex]
[tex]\[
-80i \text{ remains the same}
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
The correct product is [tex]\( 39 - 80i \)[/tex].
Thus, the answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
1. Start by using the formula for squaring a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
In our case, [tex]\( a = 8 \)[/tex] and [tex]\( b = 5i \)[/tex].
2. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
3. Calculate each term separately:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
2 \cdot 8 \cdot 5i = 80i
\][/tex]
[tex]\[
(5i)^2 = 25i^2
\][/tex]
4. Recall that [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
25i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Now, substitute these back into the equation:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
6. Combine the real parts and the imaginary parts:
[tex]\[
64 - 25 = 39
\][/tex]
[tex]\[
-80i \text{ remains the same}
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
The correct product is [tex]\( 39 - 80i \)[/tex].
Thus, the answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
