Answer :
Sure, let's break down the multiplication and simplification of [tex]\((8 - 5i)^2\)[/tex] step-by-step.
1. Expression: [tex]\((8 - 5i)^2\)[/tex]
2. Formula for Squaring a Binomial: When you square a binomial [tex]\( (a + bi)^2 \)[/tex], you use the formula:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
3. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex].
4. Square [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\( a^2 = 8^2 = 64 \)[/tex]
- [tex]\( b^2 = (-5)^2 = 25 \)[/tex]
5. Multiply [tex]\(2abi\)[/tex]:
- [tex]\( 2 \cdot 8 \cdot (-5)i = 2 \cdot 8 \cdot -5 \cdot i = -80i \)[/tex]
6. Combine [tex]\( a^2 \)[/tex], [tex]\( b^2 \)[/tex], and [tex]\(2abi\)[/tex]:
- Real part: [tex]\(64 + 25 = 89\)[/tex]
- Imaginary part: [tex]\(-80i\)[/tex]
7. Final Simplified Product:
[tex]\[
(8 - 5i)^2 = 89 - 80i
\][/tex]
So, the correct result of multiplying and simplifying [tex]\((8 - 5i)^2\)[/tex] is [tex]\( \boxed{89 - 80i} \)[/tex].
1. Expression: [tex]\((8 - 5i)^2\)[/tex]
2. Formula for Squaring a Binomial: When you square a binomial [tex]\( (a + bi)^2 \)[/tex], you use the formula:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
3. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex].
4. Square [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\( a^2 = 8^2 = 64 \)[/tex]
- [tex]\( b^2 = (-5)^2 = 25 \)[/tex]
5. Multiply [tex]\(2abi\)[/tex]:
- [tex]\( 2 \cdot 8 \cdot (-5)i = 2 \cdot 8 \cdot -5 \cdot i = -80i \)[/tex]
6. Combine [tex]\( a^2 \)[/tex], [tex]\( b^2 \)[/tex], and [tex]\(2abi\)[/tex]:
- Real part: [tex]\(64 + 25 = 89\)[/tex]
- Imaginary part: [tex]\(-80i\)[/tex]
7. Final Simplified Product:
[tex]\[
(8 - 5i)^2 = 89 - 80i
\][/tex]
So, the correct result of multiplying and simplifying [tex]\((8 - 5i)^2\)[/tex] is [tex]\( \boxed{89 - 80i} \)[/tex].
