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Multiply and simplify the product: [tex](8-5i)^2[/tex].

Select the product:

A. 39
B. 89
C. [tex]39-80i[/tex]
D. [tex]89-80i[/tex]

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].

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