Express the result in the form x+iy, where x,y are real number `i=sqrt(-1)`:
(i) `(5+9i)-:(-3+4i)`
(ii) `[(sqrt(5)+(i)/(2))(sqrt(5)-2i)]-:(6+5i)`
(iii) `((1-i)(2-i)(3-i))/(1+i)`
(iv) `(1+3i)/((1-2i)^(2))`

Answer

Let's solve each part step by step to express the results in the form \( x + iy \), where \( x \) and \( y \) are real numbers.

### Part (i): \((5 + 9i) \div (-3 + 4i)\)

1. **Multiply by the conjugate**:
\[ \frac{(5 + 9i)(-3 - 4i)}{(-3 + 4i)(-3 - 4i)} \]

2. **Calculate the denominator**:
\[ (-3)^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

3. **Calculate the numerator**:
\[ (5)(-3) + (5)(-4i) + (9i)(-3) + (9i)(-4i) = -15 - 20i - 27i - 36(-1) = -15 - 47i + 36 = 21 - 47i \]

4. **Combine results**:
\[ \frac{21 - 47i}{25} = \frac{21}{25} - \frac{47}{25}i \]

5. **Final result**:
\[ x = \frac{21}{25}, \quad y = -\frac{47}{25} \]

### Part (ii): \(\left(\sqrt{5} + \frac{i}{2}\right)\left(\sqrt{5} - 2i\right) \div (6 + 5i)\)

1. **Multiply the numerators**:
\[ \left(\sqrt{5} + \frac{i}{2}\right)\left(\sqrt{5} - 2i\right) = 5 - 2\sqrt{5}i + \frac{i\sqrt{5}}{2} - 2(-1) = 5 + 2 + \left(-2\sqrt{5} + \frac{\sqrt{5}}{2}\right)i \]

2. **Combine terms**:
\[ 7 + \left(-2\sqrt{5} + \frac{\sqrt{5}}{2}\right)i = 7 - \frac{4\sqrt{5}}{2} + \frac{\sqrt{5}}{2} = 7 - \frac{3\sqrt{5}}{2}i \]

3. **Denominator**:
\[ 6 + 5i \]

4. **Multiply by the conjugate**:
\[ \frac{(7 - \frac{3\sqrt{5}}{2}i)(6 - 5i)}{(6 + 5i)(6 - 5i)} \]

5. **Calculate the denominator**:
\[ 36 + 25 = 61 \]

6. **Calculate the numerator**:
\[ 42 - 35i - 18\sqrt{5}i + \frac{15\sqrt{5}}{2} = 42 + \left(-35 - 18\sqrt{5}\right)i + \frac{15\sqrt{5}}{2} \]

7. **Final result**:
\[ \frac{42 + \left(-35 - 18\sqrt{5}\right)i}{61} \]

### Part (iii): \(\frac{(1 - i)(2 - i)(3 - i)}{1 + i}\)

1. **Calculate the numerator**:
\[ (1 - i)(2 - i) = 2 - 3i + i^2 = 2 - 3i - 1 = 1 - 3i \]
\[ (1 - 3i)(3 - i) = 3 - i - 9i + 3i^2 = 3 - 10i - 3 = -10i \]

2. **Denominator**:
\[ 1 + i \]

3. **Multiply by the conjugate**:
\[ \frac{-10i(1 - i)}{(1 + i)(1 - i)} = \frac{-10i + 10}{1 + 1} = \frac{10 - 10i}{2} = 5 - 5i \]

### Part (iv): \(\frac{1 + 3i}{(1 - 2i)^2}\)

1. **Calculate the denominator**:
\[ (1 - 2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 = -3 - 4i \]

2. **Multiply by the conjugate**:
\[ \frac{(1 + 3i)(-3 + 4i)}{(-3 - 4i)(-3 + 4i)} = \frac{-3 + 4i - 9i + 12i^2}{9 + 16} = \frac{-3 - 12 + (-5)i}{25} = \frac{-15 - 5i}{25} \]

3. **Final result**:
\[ -\frac{3}{5} - \frac{1}{5}i \]

### Summary of Results:
1. Part (i): \( \frac{21}{25} - \frac{47}{25}i \)
2. Part (ii): \( \frac{42}{61} + \left(-\frac{35 + 18\sqrt{5}}{61}\right)i \)
3. Part (iii): \( 5 - 5i \)
4. Part (iv): \( -\frac{3}{5} - \frac{1}{5}i \)