Question

1. (1 + 8i)(-7 - 3i)

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first complex number by each term in the second complex number:
$$(1 + 8i)(-7 - 3i)=1\times(-7)+1\times(-3i)+8i\times(-7)+8i\times(-3i)$$

Step2: Simplify each product

Simplify each term:

• \(1\times(-7)= -7\)
• \(1\times(-3i)= -3i\)
• \(8i\times(-7)= -56i\)
• \(8i\times(-3i)= -24i^{2}\) (remember \(i^{2} = - 1\))

So now we have: \(-7-3i - 56i-24i^{2}\)

Step3: Substitute \(i^{2}=-1\) and combine like terms

Substitute \(i^{2}=-1\) into the expression:
\(-7-3i - 56i-24\times(-1)\)
\(=-7-3i - 56i + 24\)
Combine the real parts (\(-7 + 24\)) and the imaginary parts (\(-3i-56i\)):
Real parts: \(-7 + 24=17\)
Imaginary parts: \(-3i-56i=-59i\)
So the result is \(17-59i\)

Answer:

\(17 - 59i\)