Question

1. simplify the following: (7i - (3i + 10) + 5i)
2. multiply the following: ((2i - 5)(3i + 1))

Explanation:

Problem 12: Simplify \( 7i - (3i + 10) + 5i \)

Step 1: Remove the parentheses

Distribute the negative sign to the terms inside the parentheses: \( 7i - 3i - 10 + 5i \)

Step 2: Combine like terms (the \( i \)-terms)

Combine the \( i \)-terms: \( (7i - 3i + 5i) - 10 \)
Calculate the sum of the \( i \)-terms: \( 7i - 3i = 4i \), then \( 4i + 5i = 9i \)
So we have \( 9i - 10 \) or \( -10 + 9i \)

Step 1: Use the distributive property (FOIL method)

Multiply each term in the first binomial by each term in the second binomial:
\( 2i \times 3i + 2i \times 1 - 5 \times 3i - 5 \times 1 \)

Step 2: Simplify each term

Recall that \( i^2 = -1 \):

• \( 2i \times 3i = 6i^2 = 6(-1) = -6 \)
• \( 2i \times 1 = 2i \)
• \( -5 \times 3i = -15i \)
• \( -5 \times 1 = -5 \)

So the expression becomes: \( -6 + 2i - 15i - 5 \)

Step 3: Combine like terms

Combine the constant terms and the \( i \)-terms:

• Constant terms: \( -6 - 5 = -11 \)
• \( i \)-terms: \( 2i - 15i = -13i \)

Putting it together: \( -11 - 13i \)

Answer:

\( -10 + 9i \)

Problem 13: Multiply \( (2i - 5)(3i + 1) \)