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Answer :
To find [tex]\((8 - 5i)^2\)[/tex], we need to multiply the complex number by itself and simplify the product. Let's go through the steps:
1. Identify the components:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex], as the complex number is of the form [tex]\(a + bi\)[/tex].
2. Use the formula [tex]\((a + bi)^2 = a^2 - 2ab + (bi)^2\)[/tex]:
- Remember that [tex]\((bi)^2 = (b^2)(i^2)\)[/tex], and since [tex]\(i^2 = -1\)[/tex], it simplifies to [tex]\(-b^2\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\((bi)^2 = (-5i)^2 = 25(i^2) = 25(-1) = -25\)[/tex]
- The real component becomes: [tex]\(a^2 - b^2 = 64 - 25 = 39\)[/tex]
4. Calculate the imaginary component:
- The middle term in the expansion results in the imaginary part: [tex]\(2ab = 2(8)(-5) = -80\)[/tex]
5. Combine components:
- The result is the complex number [tex]\(39 - 80i\)[/tex].
Thus, [tex]\((8 - 5i)^2\)[/tex] evaluates to [tex]\(39 - 80i\)[/tex]. The correct option is [tex]\(39 - 80i\)[/tex].
1. Identify the components:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex], as the complex number is of the form [tex]\(a + bi\)[/tex].
2. Use the formula [tex]\((a + bi)^2 = a^2 - 2ab + (bi)^2\)[/tex]:
- Remember that [tex]\((bi)^2 = (b^2)(i^2)\)[/tex], and since [tex]\(i^2 = -1\)[/tex], it simplifies to [tex]\(-b^2\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\((bi)^2 = (-5i)^2 = 25(i^2) = 25(-1) = -25\)[/tex]
- The real component becomes: [tex]\(a^2 - b^2 = 64 - 25 = 39\)[/tex]
4. Calculate the imaginary component:
- The middle term in the expansion results in the imaginary part: [tex]\(2ab = 2(8)(-5) = -80\)[/tex]
5. Combine components:
- The result is the complex number [tex]\(39 - 80i\)[/tex].
Thus, [tex]\((8 - 5i)^2\)[/tex] evaluates to [tex]\(39 - 80i\)[/tex]. The correct option is [tex]\(39 - 80i\)[/tex].
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