Math worksheet for practicing operations with complex numbers, featuring a cartoon illustration.

Let's solve each problem step by step from the worksheet titled "Adding, Subtracting, Multiplying Complex Numbers - Independent Practice".

We'll go through each problem and simplify using the rules of complex numbers. Remember:

- A complex number is in the form $ a + bi $, where $ a $ and $ b $ are real numbers, and $ i $ is the imaginary unit with $ i^2 = -1 $.
- To add/subtract: Combine like terms (real parts and imaginary parts).
- To multiply: Use distributive property (FOIL), remembering $ i^2 = -1 $.

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Problem 1: $ (4 + 8i) + (9 + 16i) $

Add real parts and imaginary parts:

$$
(4 + 9) + (8i + 16i) = 13 + 24i
$$

✔ Answer: $ 13 + 24i $

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Problem 2: $ 5i(2 + 4i) $

Distribute $ 5i $:

$$
5i \cdot 2 + 5i \cdot 4i = 10i + 20i^2
$$

Now replace $ i^2 = -1 $:

$$
10i + 20(-1) = 10i - 20 = -20 + 10i
$$

✔ Answer: $ -20 + 10i $

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Problem 3: $ (3 + 13i) - (14 + 5i) $

Subtract real and imaginary parts:

$$
(3 - 14) + (13i - 5i) = -11 + 8i
$$

✔ Answer: $ -11 + 8i $

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Problem 4: $ 6(2 + 8i) $

Distribute 6:

$$
6 \cdot 2 + 6 \cdot 8i = 12 + 48i
$$

✔ Answer: $ 12 + 48i $

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Problem 5: $ (15 - 13i) - (18 + 2i) $

Subtract:

$$
(15 - 18) + (-13i - 2i) = -3 - 15i
$$

✔ Answer: $ -3 - 15i $

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Problem 6: $ (7 + 22i) + (15 + 4i) $

Add:

$$
(7 + 15) + (22i + 4i) = 22 + 26i
$$

✔ Answer: $ 22 + 26i $

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Problem 7: $ (2 + 15i) + (5 + 5i) $

Add:

$$
(2 + 5) + (15i + 5i) = 7 + 20i
$$

✔ Answer: $ 7 + 20i $

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Problem 8: $ 5i(7 + 8i) $

Distribute:

$$
5i \cdot 7 + 5i \cdot 8i = 35i + 40i^2
$$

Replace $ i^2 = -1 $:

$$
35i + 40(-1) = 35i - 40 = -40 + 35i
$$

✔ Answer: $ -40 + 35i $

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Problem 9: $ (7 + 8i) + (8 + 4i) $

Add:

$$
(7 + 8) + (8i + 4i) = 15 + 12i
$$

✔ Answer: $ 15 + 12i $

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Problem 10: $ (8 - 17i) - (13 - 5i) $

Subtract:

$$
(8 - 13) + (-17i - (-5i)) = -5 + (-17i + 5i) = -5 - 12i
$$

✔ Answer: $ -5 - 12i $

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✔ Final Answers:

1. $ 13 + 24i $
2. $ -20 + 10i $
3. $ -11 + 8i $
4. $ 12 + 48i $
5. $ -3 - 15i $
6. $ 22 + 26i $
7. $ 7 + 20i $
8. $ -40 + 35i $
9. $ 15 + 12i $
10. $ -5 - 12i $

Let me know if you'd like these checked or explained further!