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Answer :
To multiply and simplify the product of [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Write the expression to be expanded:
[tex]\[
(8 - 5i)^2 = (8 - 5i) \times (8 - 5i)
\][/tex]
2. Apply the distributive property (FOIL method):
- First: [tex]\(8 \times 8 = 64\)[/tex]
- Outer: [tex]\(8 \times (-5i) = -40i\)[/tex]
- Inner: [tex]\(-5i \times 8 = -40i\)[/tex]
- Last: [tex]\(-5i \times (-5i) = 25i^2\)[/tex]
3. Combine the results:
[tex]\[
64 - 40i - 40i + 25i^2
\][/tex]
4. Simplify the expression:
- Combine the imaginary terms: [tex]\(-40i - 40i = -80i\)[/tex]
- Remember that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25(-1) = -25\)[/tex]
5. Add the real parts and simplify further:
[tex]\[
64 + (-25) = 39
\][/tex]
6. Write the simplified result:
[tex]\[
39 - 80i
\][/tex]
Thus, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
1. Write the expression to be expanded:
[tex]\[
(8 - 5i)^2 = (8 - 5i) \times (8 - 5i)
\][/tex]
2. Apply the distributive property (FOIL method):
- First: [tex]\(8 \times 8 = 64\)[/tex]
- Outer: [tex]\(8 \times (-5i) = -40i\)[/tex]
- Inner: [tex]\(-5i \times 8 = -40i\)[/tex]
- Last: [tex]\(-5i \times (-5i) = 25i^2\)[/tex]
3. Combine the results:
[tex]\[
64 - 40i - 40i + 25i^2
\][/tex]
4. Simplify the expression:
- Combine the imaginary terms: [tex]\(-40i - 40i = -80i\)[/tex]
- Remember that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25(-1) = -25\)[/tex]
5. Add the real parts and simplify further:
[tex]\[
64 + (-25) = 39
\][/tex]
6. Write the simplified result:
[tex]\[
39 - 80i
\][/tex]
Thus, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
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