Determine the following: 
(5 + j7)( -3 - j8).
Your answer for the real component is 
 and the answer for the imaginary component is j 
.Each answer will be a whole number.

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!

Determine the following: (5 + j7)( -3 - j8).

Learn how to multiply the complex numbers (5 + j7) and (-3 - j8) with a step-by-step solution. This problem covers key concepts of complex number multiplication, including the distributive property and simplifying using j^2 = -1. Suitable for high school students in Grades 9-12.

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