Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], we will use the formula for the square of a binomial:
[tex]\[(a - b)^2 = a^2 - 2ab + b^2\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Let's break it down step-by-step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[8^2 = 64\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[-2 \times 8 \times 5i = -80i\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\((5i)^2 = (5)^2 \times (i)^2 = 25 \times -1 = -25\]
4. Add these values together:
Combine the real parts and the imaginary parts.
\[
64 + (-25) = 39 \quad \text{(real part)}
\]
So, the expression becomes:
\[39 - 80i\]
Thus, the simplified product of \((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
The correct answer is [tex]\(39 - 80i\)[/tex].
[tex]\[(a - b)^2 = a^2 - 2ab + b^2\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Let's break it down step-by-step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[8^2 = 64\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[-2 \times 8 \times 5i = -80i\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\((5i)^2 = (5)^2 \times (i)^2 = 25 \times -1 = -25\]
4. Add these values together:
Combine the real parts and the imaginary parts.
\[
64 + (-25) = 39 \quad \text{(real part)}
\]
So, the expression becomes:
\[39 - 80i\]
Thus, the simplified product of \((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
The correct answer is [tex]\(39 - 80i\)[/tex].
