In the following, perform the indicated operations and write the result in the form x+iy:
(i) `(5+4i)+(5-4i)`
(ii) `-2i+((3)/(2)-4i)`
(iii) `((1)/(5)+(2)/(5)i)-(4+(5)/(2)i)`
(iv) `3(7+7i)+i(7+7i)`.

Answer

Let's solve each part step by step, ensuring we express the results in the form \( x + iy \).

### (i) \( (5 + 4i) + (5 - 4i) \)

1. **Add the real parts**:
\[ 5 + 5 = 10 \]

2. **Add the imaginary parts**:
\[ 4i - 4i = 0i \]

3. **Combine the results**:
\[ 10 + 0i \]

**Final Answer**: \( 10 + 0i \)

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### (ii) \( -2i + \left(\frac{3}{2} - 4i\right) \)

1. **Identify the real part**:
\[ \text{Real part} = \frac{3}{2} \]

2. **Combine the imaginary parts**:
\[ -2i - 4i = -6i \]

3. **Combine the results**:
\[ \frac{3}{2} - 6i \]

**Final Answer**: \( \frac{3}{2} - 6i \)

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### (iii) \( \left(\frac{1}{5} + \frac{2}{5}i\right) - \left(4 + \frac{5}{2}i\right) \)

1. **Identify the real parts**:
\[ \frac{1}{5} - 4 = \frac{1}{5} - \frac{20}{5} = \frac{1 - 20}{5} = -\frac{19}{5} \]

2. **Combine the imaginary parts**:
\[ \frac{2}{5}i - \frac{5}{2}i = \frac{2}{5} - \frac{25}{10} = \frac{4 - 25}{10} = -\frac{21}{10}i \]

3. **Combine the results**:
\[ -\frac{19}{5} - \frac{21}{10}i \]

**Final Answer**: \( -\frac{19}{5} - \frac{21}{10}i \)

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### (iv) \( 3(7 + 7i) + i(7 + 7i) \)

1. **Distribute \( 3 \)**:
\[ 3 \times 7 + 3 \times 7i = 21 + 21i \]

2. **Distribute \( i \)**:
\[ i \times 7 + i \times 7i = 7i + 7i^2 \]
Since \( i^2 = -1 \):
\[ 7i - 7 = -7 + 7i \]

3. **Combine the results**:
\[ (21 - 7) + (21i + 7i) = 14 + 28i \]

**Final Answer**: \( 14 + 28i \)

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### Summary of Answers:
1. \( 10 + 0i \)
2. \( \frac{3}{2} - 6i \)
3. \( -\frac{19}{5} - \frac{21}{10}i \)
4. \( 14 + 28i \)

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