Answer :
To multiply and simplify the product [tex]\((8 - 5i)^2\)[/tex], we can use the formula for squaring a binomial. Here's a step-by-step breakdown of how to do it:
1. Understand the expression: We need to find the square of [tex]\((8 - 5i)\)[/tex]. This means we need to multiply the expression by itself:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
2. Apply the formula for squaring a complex number:
[tex]\[
(a - bi)^2 = a^2 - 2ab \cdot i + (bi)^2
\][/tex]
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate the real part:
- First, calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- Then, calculate [tex]\(b^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
- Subtract [tex]\(b^2\)[/tex] from [tex]\(a^2\)[/tex]:
[tex]\[
64 - 25 = 39
\][/tex]
4. Calculate the imaginary part:
- Compute [tex]\(2ab\)[/tex]:
[tex]\[
2 \times 8 \times 5 = 80
\][/tex]
- Since the original expression is [tex]\(8 - 5i\)[/tex], the imaginary part will be negative due to [tex]\(-2ab\)[/tex], giving us [tex]\(-80i\)[/tex].
5. Combine the real and imaginary parts:
- The result is a complex number with the real part and the negative imaginary part, yielding:
[tex]\[
39 - 80i
\][/tex]
Therefore, the simplified form of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct choice from the options provided is [tex]\(39 - 80i\)[/tex].
1. Understand the expression: We need to find the square of [tex]\((8 - 5i)\)[/tex]. This means we need to multiply the expression by itself:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
2. Apply the formula for squaring a complex number:
[tex]\[
(a - bi)^2 = a^2 - 2ab \cdot i + (bi)^2
\][/tex]
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate the real part:
- First, calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- Then, calculate [tex]\(b^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
- Subtract [tex]\(b^2\)[/tex] from [tex]\(a^2\)[/tex]:
[tex]\[
64 - 25 = 39
\][/tex]
4. Calculate the imaginary part:
- Compute [tex]\(2ab\)[/tex]:
[tex]\[
2 \times 8 \times 5 = 80
\][/tex]
- Since the original expression is [tex]\(8 - 5i\)[/tex], the imaginary part will be negative due to [tex]\(-2ab\)[/tex], giving us [tex]\(-80i\)[/tex].
5. Combine the real and imaginary parts:
- The result is a complex number with the real part and the negative imaginary part, yielding:
[tex]\[
39 - 80i
\][/tex]
Therefore, the simplified form of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct choice from the options provided is [tex]\(39 - 80i\)[/tex].
