Answer :
Sure, I'll show you how to multiply and simplify [tex]\((8 - 5i)^2\)[/tex] step-by-step.
Step 1: Write down the expression.
[tex]\[
(8 - 5i)^2
\][/tex]
Step 2: Use the formula for the square of a binomial [tex]\((a - b)^2\)[/tex]:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
Step 3: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula.
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot (5i) + (5i)^2
\][/tex]
Step 4: Calculate each term:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
-2 \cdot 8 \cdot 5i = -80i
\][/tex]
[tex]\[
(5i)^2 = 25i^2
\][/tex]
Step 5: Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
Step 6: Combine all the terms:
[tex]\[
64 - 80i - 25
\][/tex]
Step 7: Simplify by combining the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So, the expression simplifies to:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
Step 1: Write down the expression.
[tex]\[
(8 - 5i)^2
\][/tex]
Step 2: Use the formula for the square of a binomial [tex]\((a - b)^2\)[/tex]:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
Step 3: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula.
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot (5i) + (5i)^2
\][/tex]
Step 4: Calculate each term:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
-2 \cdot 8 \cdot 5i = -80i
\][/tex]
[tex]\[
(5i)^2 = 25i^2
\][/tex]
Step 5: Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
Step 6: Combine all the terms:
[tex]\[
64 - 80i - 25
\][/tex]
Step 7: Simplify by combining the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So, the expression simplifies to:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
