Let's solve the problem step-by-step:
1. Identify the Complex Number and its Conjugate:
The given complex number is [tex]\(-3 - 5i\)[/tex]. To find its conjugate, we change the sign of the imaginary part.
[tex]\[
\text{Conjugate of } -3 - 5i \text{ is } -3 + 5i.
\][/tex]
2. Calculate the Product:
We need to multiply the complex number by its conjugate:
[tex]\[
(-3 - 5i) \times (-3 + 5i).
\][/tex]
3. Use the Formula for Product of a Complex Number and its Conjugate:
The product of a complex number [tex]\(a + bi\)[/tex] and its conjugate [tex]\(a - bi\)[/tex] is given by:
[tex]\[
(a + bi)(a - bi) = a^2 + b^2.
\][/tex]
In our case, [tex]\(a = -3\)[/tex] and [tex]\(b = -5\)[/tex].
4. Substitute the Values:
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
[tex]\[
(-3)^2 + (-5)^2 = 9 + 25.
\][/tex]
5. Perform the Addition:
Calculate the sum:
[tex]\[
9 + 25 = 34.
\][/tex]
6. Interpret the Result:
Since the product of a complex number and its conjugate is always a real number (no imaginary part), the result [tex]\(34\)[/tex] is purely real with an imaginary part of [tex]\(0\)[/tex]. Therefore:
[tex]\[
a = 34 \quad \text{and} \quad b = 0.
\][/tex]
So, the product of [tex]\(-3-5i\)[/tex] and its conjugate is [tex]\((34.0, 0.0)\)[/tex]. Thus,
- The real number [tex]\(a\)[/tex] equals [tex]\(\boxed{34}\)[/tex],
- The real number [tex]\(b\)[/tex] equals [tex]\(\boxed{0}\)[/tex].
