Perform the indicated operation and give your anwer in the form `x+iy`, where x and y are real numbers and `i=sqrt(-1)`:
(i) `((1)/(2)+(1)/(4)i)(-(2)/(3)-(1)/(4)i)`
(ii) `(5+2i)/(-1+sqrt(3)i)`.
(iii) `(sqrt(5)-7i)(sqrt(5)-7i)^(2)+(-2+7i)^(2)`.

Answer

Let's solve the given complex number problems step by step.

### (i) \(\left(\frac{1}{2} + \frac{1}{4}i\right)\left(-\frac{2}{3} - \frac{1}{4}i\right)\)

1. **Multiply the complex numbers using the distributive property (FOIL method)**:
\[ \left(\frac{1}{2} \cdot -\frac{2}{3}\right) + \left(\frac{1}{2} \cdot -\frac{1}{4}i\right) + \left(\frac{1}{4}i \cdot -\frac{2}{3}\right) + \left(\frac{1}{4}i \cdot -\frac{1}{4}i\right) \]

2. **Calculate each term**:
- First term: \(\frac{1}{2} \cdot -\frac{2}{3} = -\frac{1}{3}\)
- Second term: \(\frac{1}{2} \cdot -\frac{1}{4}i = -\frac{1}{8}i\)
- Third term: \(\frac{1}{4}i \cdot -\frac{2}{3} = -\frac{1}{6}i\)
- Fourth term: \(\frac{1}{4}i \cdot -\frac{1}{4}i = -\frac{1}{16}i^2 = \frac{1}{16}\) (since \(i^2 = -1\))

3. **Combine the real and imaginary parts**:
- Real part: \(-\frac{1}{3} + \frac{1}{16}\)
- Imaginary part: \(-\frac{1}{8}i - \frac{1}{6}i\)

4. **Finding a common denominator for the real part**:
- LCM of 3 and 16 is 48.
- \(-\frac{1}{3} = -\frac{16}{48}\) and \(\frac{1}{16} = \frac{3}{48}\).
- Combine: \(-\frac{16}{48} + \frac{3}{48} = -\frac{13}{48}\).

5. **Finding a common denominator for the imaginary part**:
- LCM of 8 and 6 is 24.
- \(-\frac{1}{8} = -\frac{3}{24}\) and \(-\frac{1}{6} = -\frac{4}{24}\).
- Combine: \(-\frac{3}{24} - \frac{4}{24} = -\frac{7}{24}\).

6. **Final answer**:
\[ -\frac{13}{48} - \frac{7}{24}i \]