Answer :
To solve the problem of multiplying and simplifying the product of [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Understand the Formula: We want to square the expression [tex]\((8 - 5i)\)[/tex]. We can use the formula for squaring a binomial:
[tex]\[
(a - bi)^2 = a^2 - 2ab \cdot i + (bi)^2
\][/tex]
2. Substitute the Values:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate the Real Part:
- Use the formula [tex]\(a^2 - b^2\)[/tex].
- Calculate [tex]\(8^2 = 64\)[/tex].
- Calculate [tex]\((5i)^2 = 25i^2 = 25(-1) = -25\)[/tex].
- The real part is [tex]\(64 - 25 = 39\)[/tex].
4. Calculate the Imaginary Part:
- Use the formula [tex]\(-2ab \cdot i\)[/tex].
- Calculate [tex]\(-2 \times 8 \times 5 = -80\)[/tex].
- The imaginary part is [tex]\(-80i\)[/tex].
5. Combine the Parts:
- Putting it all together, the expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Thus, the select product is [tex]\(\boxed{39 - 80i}\)[/tex].
1. Understand the Formula: We want to square the expression [tex]\((8 - 5i)\)[/tex]. We can use the formula for squaring a binomial:
[tex]\[
(a - bi)^2 = a^2 - 2ab \cdot i + (bi)^2
\][/tex]
2. Substitute the Values:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate the Real Part:
- Use the formula [tex]\(a^2 - b^2\)[/tex].
- Calculate [tex]\(8^2 = 64\)[/tex].
- Calculate [tex]\((5i)^2 = 25i^2 = 25(-1) = -25\)[/tex].
- The real part is [tex]\(64 - 25 = 39\)[/tex].
4. Calculate the Imaginary Part:
- Use the formula [tex]\(-2ab \cdot i\)[/tex].
- Calculate [tex]\(-2 \times 8 \times 5 = -80\)[/tex].
- The imaginary part is [tex]\(-80i\)[/tex].
5. Combine the Parts:
- Putting it all together, the expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Thus, the select product is [tex]\(\boxed{39 - 80i}\)[/tex].
