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Chapter 1: Problem 14

Find each product and write the result in standard form. $$ (8-4 i)(-3+9 i) $$

Short Answer

Expert verified
The product of the complex numbers \( (8 - 4i) \) and \( (-3 + 9i) \) is \( 12 + 84i \).

Step by step solution

01

Distribute

The first step involves using the distributive property to multiply each term in the first complex number with each term in the second complex number. This yields the equation: \( 8 \cdot (-3) + 8 \cdot 9i - 4i \cdot (-3) - 4i \cdot 9i \). After performing these multiplications, the expression simplifies to \( -24 + 72i + 12i - 36i^2 \).
02

Replace \( i^2 \)

The term \( i^2 \) can be replaced with -1 (since \( i^2 = -1 \)). So the expression becomes \( -24 + 72i + 12i + 36 \).
03

Combine Like Terms

Combine the real parts and the imaginary parts separately. This results in \( (-24 + 36) + (72i + 12i) \), which simplifies to \( 12 + 84i \).

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