Problem 13 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (7-5 i)(-2-3 i) $$

Step-by-Step Solution

Answer

The product of \( (7-5i) \) and \( (-2-3i) \) is \( 1-11i \)

Step 1: Distribute the terms

First, distribute both terms in \( (7-5i) \) to each term in \( (-2-3i) \). This gives: \( (7)(-2) + (7)(-3i) - (5i)(-2) - (5i)(-3i) \)

Step 2: Simplify the terms

Next, simplify each of the terms, using the fact that \( i^{2} = -1 \): \( -14 - 21i +10i -15i^{2} \).

Step 3: Combine like terms

Combine the real and imaginary terms: \( -14 - 11i -15(-1) \).

Step 4: Simplify the expression

Now use the fact that \( i^{2} = -1 \) to simplify our expression to its final form: \( -14 + 15 - 11i = 1 - 11i \).