Simplify each of the following and express it in the form a + ib :
`{:((i),2(3+4i)+i(5-6i),(ii),(3+sqrt(-16))-(4-sqrt(-9))),((iii),(-5+6i)-(2+i),(iv),(8-4i)-(-3+5i)),((v),(1-i)^(2)(1+i)-(3-4i)^(2),(vi),(5+sqrt(-3))(5-sqrt(-3))),((vii),(3+4i)(2-3i),(viii),(-2+sqrt(-3))(-3+2 sqrt(-3))):}`

Answer

Let's simplify each of the given expressions step by step and express them in the form \( a + ib \).

### (i) \( 2(3 + 4i) + i(5 - 6i) \)

1. Distribute the terms:
\[ 2(3 + 4i) = 6 + 8i \]
\[ i(5 - 6i) = 5i - 6i^2 = 5i + 6 \quad (\text{since } i^2 = -1) \]

2. Combine the results:
\[ 6 + 8i + 5i + 6 = 12 + 13i \]

**Final Answer: \( 12 + 13i \)**

### (ii) \( (3 + \sqrt{-16}) - (4 - \sqrt{-9}) \)

1. Rewrite the square roots:
\[ \sqrt{-16} = 4i \quad \text{and} \quad \sqrt{-9} = 3i \]

2. Substitute and simplify:
\[ (3 + 4i) - (4 - 3i) = 3 + 4i - 4 + 3i = (3 - 4) + (4i + 3i) = -1 + 7i \]

**Final Answer: \( -1 + 7i \)**

### (iii) \( (-5 + 6i) - (2 + i) \)

1. Distribute the negative sign:
\[ -5 + 6i - 2 - i = (-5 - 2) + (6i - i) = -7 + 5i \]

**Final Answer: \( -7 + 5i \)**

### (iv) \( (8 - 4i) - (-3 + 5i) \)

1. Distribute the negative sign:
\[ 8 - 4i + 3 - 5i = (8 + 3) + (-4i - 5i) = 11 - 9i \]

**Final Answer: \( 11 - 9i \)**

### (v) \( (1 - i)^2(1 + i) - (3 - 4i)^2 \)

1. Calculate \( (1 - i)^2 \):
\[ (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \]

2. Calculate \( -2i(1 + i) \):
\[ -2i(1 + i) = -2i - 2i^2 = -2i + 2 = 2 - 2i \]

3. Calculate \( (3 - 4i)^2 \):
\[ (3 - 4i)^2 = 9 - 24i + 16i^2 = 9 - 24i - 16 = -7 - 24i \]

4. Combine:
\[ (2 - 2i) - (-7 - 24i) = 2 - 2i + 7 + 24i = (2 + 7) + (-2i + 24i) = 9 + 22i \]

**Final Answer: \( 9 + 22i \)**

### (vi) \( (5 + \sqrt{-3})(5 - \sqrt{-3}) \)

1. Rewrite the square roots:
\[ \sqrt{-3} = i\sqrt{3} \]

2. Use the difference of squares:
\[ (5 + i\sqrt{3})(5 - i\sqrt{3}) = 5^2 - (i\sqrt{3})^2 = 25 - (-3) = 25 + 3 = 28 \]

**Final Answer: \( 28 + 0i \) (or simply \( 28 \))**

### (vii) \( (3 + 4i)(2 - 3i) \)

1. Distribute:
\[ 3 \cdot 2 + 3 \cdot (-3i) + 4i \cdot 2 + 4i \cdot (-3i) = 6 - 9i + 8i - 12i^2 \]

2. Combine:
\[ 6 - 9i + 8i + 12 = (6 + 12) + (-9i + 8i) = 18 - i \]

**Final Answer: \( 18 - i \)**

### (viii) \( (-2 + \sqrt{-3})(-3 + 2\sqrt{-3}) \)

1. Rewrite the square roots:
\[ \sqrt{-3} = i\sqrt{3} \]

2. Substitute and distribute:
\[ (-2 + i\sqrt{3})(-3 + 2i\sqrt{3}) = (-2)(-3) + (-2)(2i\sqrt{3}) + (i\sqrt{3})(-3) + (i\sqrt{3})(2i\sqrt{3}) \]
\[ = 6 - 4i\sqrt{3} - 3i\sqrt{3} + 2(-3) = 6 - 4i\sqrt{3} - 3i\sqrt{3} - 6 \]

3. Combine:
\[ 0 - (4i\sqrt{3} + 3i\sqrt{3}) = -7i\sqrt{3} \]

**Final Answer: \( 0 - 7i\sqrt{3} \) (or simply \( -7i\sqrt{3} \))**