We are given the complex numbers $(5 + j7)$ and $(-3 - j8)$. We will determine their product by expanding using the distributive property:

$(5 + j7)(-3 - j8)$

Step 1: Expand Using Distribution

$5(-3) + 5(-j8) + j7(-3) + j7(-j8)$

$-15 - j40 - j21 + j^2(56)$

Step 2: Simplify Using $j^2 = -1$

Since $j^2 = -1$, we replace $j^2(56)$ with $-56$:

$-15 - j40 - j21 - 56$

$(-15 - 56) + (-j40 - j21)$

$-71 - j61$

Final Answer:

• Real Component: -71
• Imaginary Component: -61 (or $j61$ in the requested format)

Do you need further clarification?

Related Problems:

1. Compute $(4 + j6)(-2 + j3)$.
2. Find the modulus of $5 + j7$.
3. Compute the conjugate of $-3 - j8$.
4. Simplify $(2 + j5)^2$.
5. Determine the argument (angle) of $-3 - j8$.

Tip: Always apply $j^2 = -1$ to simplify complex number multiplication!