Question

topic: multiplying and dividing complex numbers - worksheet 2
simplify:

1. ( 4(2 - 3i) + 6i ) 2. ( (7 - 4i)(4 - i) )
2. ( (5 - 3i)^2 ) 4. ( 8i + 5(4 - 1i) )
3. ( 9i + 9(5 - 3i) ) 6. ( (5 + 9i)(5 - 9i) )
4. ( (7 - 6i)(5 - i) ) 8. ( (9 - 5i)(7 + 4i) )
5. ( (4 - 6i)^2 ) 10. ( (6 - 3i)(-2 + i) )

Explanation:

Problem 1: Simplify \( 4(2 - 3i) + 6i \)

Step 1: Distribute the 4

Use the distributive property \( a(b + c) = ab + ac \) to expand \( 4(2 - 3i) \).
\( 4(2 - 3i) = 4 \times 2 - 4 \times 3i = 8 - 12i \)

Step 2: Combine like terms

Add \( 6i \) to the result from Step 1.
\( 8 - 12i + 6i = 8 - 6i \)

Step 1: Use the FOIL method

Multiply the First, Outer, Inner, and Last terms:

• First: \( 7 \times 4 = 28 \)
• Outer: \( 7 \times (-i) = -7i \)
• Inner: \( -4i \times 4 = -16i \)
• Last: \( -4i \times (-i) = 4i^2 \) (remember \( i^2 = -1 \))

So, \( (7 - 4i)(4 - i) = 28 - 7i - 16i + 4i^2 \)

Step 2: Simplify \( i^2 \) and combine like terms

Substitute \( i^2 = -1 \):
\( 28 - 7i - 16i + 4(-1) = 28 - 23i - 4 \)
Combine the constant terms:
\( 28 - 4 - 23i = 24 - 23i \)

Step 1: Expand using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \)

Here, \( a = 5 \) and \( b = 3i \).
\( (5 - 3i)^2 = 5^2 - 2 \times 5 \times 3i + (3i)^2 \)

Step 2: Simplify each term

• \( 5^2 = 25 \)
• \( -2 \times 5 \times 3i = -30i \)
• \( (3i)^2 = 9i^2 = 9(-1) = -9 \) (since \( i^2 = -1 \))

Step 3: Combine like terms

\( 25 - 30i - 9 = 16 - 30i \)

Answer:

\( 8 - 6i \)

Problem 2: Simplify \( (7 - 4i)(4 - i) \)