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Answer :
Sure! Let's find the product of [tex]\((8 - 5i)^2\)[/tex] step by step.
1. Write down the expression: We want to compute [tex]\((8 - 5i)^2\)[/tex].
2. Expand the square using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
3. Calculate each term separately:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(2 \cdot 8 \cdot 5i = 80i\)[/tex]
- [tex]\((5i)^2 = 5^2 \cdot i^2 = 25 \cdot i^2\)[/tex]
4. Remember that [tex]\(i^2 = -1\)[/tex], so:
- [tex]\((5i)^2 = 25 \cdot (-1) = -25\)[/tex]
5. Substitute these values back into the expanded form:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
6. Combine the real parts:
- [tex]\(64 - 25 = 39\)[/tex]
7. Combine all parts to get the final result:
- [tex]\((8 - 5i)^2 = 39 - 80i\)[/tex]
So, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{39 - 80i} \][/tex]
1. Write down the expression: We want to compute [tex]\((8 - 5i)^2\)[/tex].
2. Expand the square using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
3. Calculate each term separately:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(2 \cdot 8 \cdot 5i = 80i\)[/tex]
- [tex]\((5i)^2 = 5^2 \cdot i^2 = 25 \cdot i^2\)[/tex]
4. Remember that [tex]\(i^2 = -1\)[/tex], so:
- [tex]\((5i)^2 = 25 \cdot (-1) = -25\)[/tex]
5. Substitute these values back into the expanded form:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
6. Combine the real parts:
- [tex]\(64 - 25 = 39\)[/tex]
7. Combine all parts to get the final result:
- [tex]\((8 - 5i)^2 = 39 - 80i\)[/tex]
So, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{39 - 80i} \][/tex]
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