Answer :
To solve the expression [tex]\(\frac{-30 - 10i}{-8 - 6i}\)[/tex] and write the answer in standard form, which is [tex]\(a + bi\)[/tex], let's go through the steps:
1. Identify the Expression: We need to divide two complex numbers: the numerator is [tex]\(-30 - 10i\)[/tex] and the denominator is [tex]\(-8 - 6i\)[/tex].
2. Multiply by the Conjugate: To simplify the division of complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(-8 - 6i\)[/tex] is [tex]\(-8 + 6i\)[/tex].
3. Perform the Multiplications:
- Multiply the numerator by the conjugate:
[tex]\[
(-30 - 10i)(-8 + 6i) = (-30)(-8) + (-30)(6i) + (-10i)(-8) + (-10i)(6i)
\][/tex]
Simplify each term:
[tex]\[
= 240 + (-180i) + 80i + 60i^2
\][/tex]
Substitute [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 240 - 180i + 80i - 60
\][/tex]
Combine like terms:
[tex]\[
= (240 - 60) + (-180i + 80i)
\][/tex]
[tex]\[
= 180 - 100i
\][/tex]
- Multiply the denominator by its conjugate:
[tex]\[
(-8 - 6i)(-8 + 6i) = (-8)^2 + (-6i)(6i)
\][/tex]
Simplify:
[tex]\[
= 64 + 36i^2
\][/tex]
Substitute [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 64 - 36
\][/tex]
[tex]\[
= 100
\][/tex]
4. Divide to Simplify: Now divide the result from the numerator by the result from the denominator:
[tex]\[
\frac{180 - 100i}{100} = \frac{180}{100} - \frac{100i}{100}
\][/tex]
Simplify each part:
[tex]\[
= 1.8 - i
\][/tex]
5. Write the Answer in Standard Form: The result of the division [tex]\(\frac{-30 - 10i}{-8 - 6i}\)[/tex] in standard form is:
[tex]\[
1.8 - i
\][/tex]
Thus, the solution to the operation is [tex]\((3.0, -1.0)\)[/tex], which means the real part is 3.0 and the imaginary part is [tex]\(-1.0\)[/tex]. In standard form, it should be written as: [tex]\(3.0 - 1.0i\)[/tex].
1. Identify the Expression: We need to divide two complex numbers: the numerator is [tex]\(-30 - 10i\)[/tex] and the denominator is [tex]\(-8 - 6i\)[/tex].
2. Multiply by the Conjugate: To simplify the division of complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(-8 - 6i\)[/tex] is [tex]\(-8 + 6i\)[/tex].
3. Perform the Multiplications:
- Multiply the numerator by the conjugate:
[tex]\[
(-30 - 10i)(-8 + 6i) = (-30)(-8) + (-30)(6i) + (-10i)(-8) + (-10i)(6i)
\][/tex]
Simplify each term:
[tex]\[
= 240 + (-180i) + 80i + 60i^2
\][/tex]
Substitute [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 240 - 180i + 80i - 60
\][/tex]
Combine like terms:
[tex]\[
= (240 - 60) + (-180i + 80i)
\][/tex]
[tex]\[
= 180 - 100i
\][/tex]
- Multiply the denominator by its conjugate:
[tex]\[
(-8 - 6i)(-8 + 6i) = (-8)^2 + (-6i)(6i)
\][/tex]
Simplify:
[tex]\[
= 64 + 36i^2
\][/tex]
Substitute [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 64 - 36
\][/tex]
[tex]\[
= 100
\][/tex]
4. Divide to Simplify: Now divide the result from the numerator by the result from the denominator:
[tex]\[
\frac{180 - 100i}{100} = \frac{180}{100} - \frac{100i}{100}
\][/tex]
Simplify each part:
[tex]\[
= 1.8 - i
\][/tex]
5. Write the Answer in Standard Form: The result of the division [tex]\(\frac{-30 - 10i}{-8 - 6i}\)[/tex] in standard form is:
[tex]\[
1.8 - i
\][/tex]
Thus, the solution to the operation is [tex]\((3.0, -1.0)\)[/tex], which means the real part is 3.0 and the imaginary part is [tex]\(-1.0\)[/tex]. In standard form, it should be written as: [tex]\(3.0 - 1.0i\)[/tex].
