Simplifying Complex Numbers worksheet with problems and color-coded sections.

To solve the problem of simplifying complex numbers in the given image, we will go through each expression step by step. Let's break it down systematically.

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Step 1: Simplify Each Expression

We will simplify each complex number expression one by one. Recall that:
- $ i^2 = -1 $
- Addition and subtraction of complex numbers involve combining real parts and imaginary parts separately.
- Multiplication involves distributing terms and using $ i^2 = -1 $.

---

#### Row 1:
1. A: $ -4i - 2 - 2i $
Combine like terms:
\[
-4i - 2i - 2 = -6i - 2
\]
Simplified: $ -2 - 6i $

2. B: $ 7i + 5i $
Combine like terms:
\[
7i + 5i = 12i
\]
Simplified: $ 0 + 12i $

3. C: $ -5 - 4i - 3i $
Combine like terms:
\[
-5 - 4i - 3i = -5 - 7i
\]
Simplified: $ -5 - 7i $

4. D: $ -7 - (7 + 4i) $
Distribute the negative sign:
\[
-7 - 7 - 4i = -14 - 4i
\]
Simplified: $ -14 - 4i $

5. E: $ 6i - 5 + 5i $
Combine like terms:
\[
6i + 5i - 5 = 11i - 5
\]
Simplified: $ -5 + 11i $

---

#### Row 2:
6. F: $ -i + 3 - 2i $
Combine like terms:
\[
-i - 2i + 3 = -3i + 3
\]
Simplified: $ 3 - 3i $

7. G: $ i - (3 + 3i) $
Distribute the negative sign:
\[
i - 3 - 3i = -3 - 2i
\]
Simplified: $ -3 - 2i $

8. H: $ -7 - (-8 + 3i) $
Distribute the negative sign:
\[
-7 + 8 - 3i = 1 - 3i
\]
Simplified: $ 1 - 3i $

9. I: $ (-7 - i) - (3 + 6i) $
Distribute and combine like terms:
\[
-7 - i - 3 - 6i = -10 - 7i
\]
Simplified: $ -10 - 7i $

10. J: $ (3 + 4i) - (-7 - 3i) $
Distribute and combine like terms:
\[
3 + 4i + 7 + 3i = 10 + 7i
\]
Simplified: $ 10 + 7i $

---

#### Row 3:
11. K: $ 5 + (1 + 5i) - i $
Combine like terms:
\[
5 + 1 + 5i - i = 6 + 4i
\]
Simplified: $ 6 + 4i $

12. L: $ (-7 + 7i) + (5 - 8i) $
Combine like terms:
\[
-7 + 7i + 5 - 8i = -2 - i
\]
Simplified: $ -2 - i $

13. M: $ (-9 + 4i)(4 - 2i) $
Use the distributive property (FOIL):
\[
(-9)(4) + (-9)(-2i) + (4i)(4) + (4i)(-2i)
\]
\[
= -36 + 18i + 16i - 8i^2
\]
Since $ i^2 = -1 $:
\[
= -36 + 18i + 16i - 8(-1)
\]
\[
= -36 + 34i + 8
\]
\[
= -28 + 34i
\]
Simplified: $ -28 + 34i $

14. N: $ (-5 + 5i)(5 - 4i) $
Use the distributive property (FOIL):
\[
(-5)(5) + (-5)(-4i) + (5i)(5) + (5i)(-4i)
\]
\[
= -25 + 20i + 25i - 20i^2
\]
Since $ i^2 = -1 $:
\[
= -25 + 20i + 25i - 20(-1)
\]
\[
= -25 + 45i + 20
\]
\[
= -5 + 45i
\]
Simplified: $ -5 + 45i $

15. O: $ (8 - 8i)(2 - 6i) $
Use the distributive property (FOIL):
\[
(8)(2) + (8)(-6i) + (-8i)(2) + (-8i)(-6i)
\]
\[
= 16 - 48i - 16i + 48i^2
\]
Since $ i^2 = -1 $:
\[
= 16 - 48i - 16i + 48(-1)
\]
\[
= 16 - 64i - 48
\]
\[
= -32 - 64i
\]
Simplified: $ -32 - 64i $

---

#### Row 4:
16. P: $ -5 - 2i(2 - 3i) $
Distribute $ -2i $:
\[
-5 - 2i(2) + 2i(3i)
\]
\[
= -5 - 4i + 6i^2
\]
Since $ i^2 = -1 $:
\[
= -5 - 4i + 6(-1)
\]
\[
= -5 - 4i - 6
\]
\[
= -11 - 4i
\]
Simplified: $ -11 - 4i $

17. Q: $ \frac{-3 + 6i}{1 - 6i} $
Multiply numerator and denominator by the conjugate of the denominator ($ 1 + 6i $):
\[
\frac{(-3 + 6i)(1 + 6i)}{(1 - 6i)(1 + 6i)}
\]
Simplify the denominator:
\[
(1 - 6i)(1 + 6i) = 1^2 - (6i)^2 = 1 - 36i^2 = 1 - 36(-1) = 1 + 36 = 37
\]
Simplify the numerator:
\[
(-3 + 6i)(1 + 6i) = (-3)(1) + (-3)(6i) + (6i)(1) + (6i)(6i)
\]
\[
= -3 - 18i + 6i + 36i^2
\]
Since $ i^2 = -1 $:
\[
= -3 - 18i + 6i + 36(-1)
\]
\[
= -3 - 12i - 36
\]
\[
= -39 - 12i
\]
So:
\[
\frac{-39 - 12i}{37} = -\frac{39}{37} - \frac{12}{37}i
\]
Simplified: $ -\frac{39}{37} - \frac{12}{37}i $

18. R: $ (6 + 6i)^2 $
Use the formula $ (a + b)^2 = a^2 + 2ab + b^2 $:
\[
(6 + 6i)^2 = 6^2 + 2(6)(6i) + (6i)^2
\]
\[
= 36 + 72i + 36i^2
\]
Since $ i^2 = -1 $:
\[
= 36 + 72i + 36(-1)
\]
\[
= 36 + 72i - 36
\]
\[
= 72i
\]
Simplified: $ 0 + 72i $

19. S: $ (4 - 3i)^2 $
Use the formula $ (a - b)^2 = a^2 - 2ab + b^2 $:
\[
(4 - 3i)^2 = 4^2 - 2(4)(3i) + (3i)^2
\]
\[
= 16 - 24i + 9i^2
\]
Since $ i^2 = -1 $:
\[
= 16 - 24i + 9(-1)
\]
\[
= 16 - 24i - 9
\]
\[
= 7 - 24i
\]
Simplified: $ 7 - 24i $

20. T: $ \frac{3i}{1 + 7i} $
Multiply numerator and denominator by the conjugate of the denominator ($ 1 - 7i $):
\[
\frac{3i(1 - 7i)}{(1 + 7i)(1 - 7i)}
\]
Simplify the denominator:
\[
(1 + 7i)(1 - 7i) = 1^2 - (7i)^2 = 1 - 49i^2 = 1 - 49(-1) = 1 + 49 = 50
\]
Simplify the numerator:
\[
3i(1 - 7i) = 3i - 21i^2
\]
Since $ i^2 = -1 $:
\[
= 3i - 21(-1)
\]
\[
= 3i + 21
\]
So:
\[
\frac{21 + 3i}{50} = \frac{21}{50} + \frac{3}{50}i
\]
Simplified: $ \frac{21}{50} + \frac{3}{50}i $

---

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|c|c|}
\hline
A & B & C & D & E \\
\hline
-2 - 6i & 0 + 12i & -5 - 7i & -14 - 4i & -5 + 11i \\
\hline
F & G & H & I & J \\
\hline
3 - 3i & -3 - 2i & 1 - 3i & -10 - 7i & 10 + 7i \\
\hline
K & L & M & N & O \\
\hline
6 + 4i & -2 - i & -28 + 34i & -5 + 45i & -32 - 64i \\
\hline
P & Q & R & S & T \\
\hline
-11 - 4i & -\frac{39}{37} - \frac{12}{37}i & 0 + 72i & 7 - 24i & \frac{21}{50} + \frac{3}{50}i \\
\hline
\end{array}
}
\]