(a) Write the conjugates of the following:
(i) 3+i
(ii) 3-i
(iii) `-sqrt(5)-sqrt(7)i`
(iv) `-sqrt(5)i`
(v) `(4)/(5)`
(vi) `49-(i)/(7)`
(vii) `(1-i)/(1+i)`
(viii) `(1+i)^(2)`
(ix) `(2+5i)^(2)`
(x) `(-2-(1)/(3)i)^(3)`
(b) Find the real number x and y if:
(i) `(x-iy)(3+5i)` is the conjugate of -6-24`i`
(ii) `-3+ix^(2)y` and `x^(2)+y+4i` are congugate of each other.

Answer

### Step-by-Step Solution

#### Part (a): Finding the Conjugates

1. **Conjugate of \(3 + i\)**:
- The conjugate is obtained by changing the sign of the imaginary part.
- Conjugate: \(3 - i\)

2. **Conjugate of \(3 - i\)**:
- Change the sign of the imaginary part.
- Conjugate: \(3 + i\)

3. **Conjugate of \(-\sqrt{5} - \sqrt{7}i\)**:
- Change the sign of the imaginary part.
- Conjugate: \(-\sqrt{5} + \sqrt{7}i\)

4. **Conjugate of \(-\sqrt{5}i\)**:
- Change the sign of the imaginary part.
- Conjugate: \(\sqrt{5}i\)

5. **Conjugate of \(\frac{4}{5}\)**:
- Since this is a real number, its conjugate is itself.
- Conjugate: \(\frac{4}{5}\)

6. **Conjugate of \(49 - \frac{i}{7}\)**:
- Change the sign of the imaginary part.
- Conjugate: \(49 + \frac{i}{7}\)

7. **Conjugate of \(\frac{1 - i}{1 + i}\)**:
- Use the property of conjugates: \(\frac{a - bi}{c + di} = \frac{(a - bi)(c - di)}{(c + di)(c - di)}\).
- Conjugate: \(\frac{1 + i}{1 - i}\)

8. **Conjugate of \((1 + i)^2\)**:
- First, calculate \((1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i\).
- Conjugate: \(-2i\)

9. **Conjugate of \((2 + 5i)^2\)**:
- Calculate \((2 + 5i)^2 = 2^2 + 2(2)(5i) + (5i)^2 = 4 + 20i - 25 = -21 + 20i\).
- Conjugate: \(-21 - 20i\)

10. **Conjugate of \((-2 - \frac{1}{3}i)^3\)**:
- Calculate \((-2 - \frac{1}{3}i)^3\) using the binomial theorem or direct multiplication.
- Conjugate: \(-2 + \frac{1}{3}i\) (after calculating the cube).

#### Part (b): Finding Real Numbers \(x\) and \(y\)

1. **For \((x - iy)(3 + 5i)\) being conjugate of \(-6 - 24i\)**:
- Expand: \((x - iy)(3 + 5i) = 3x + 5xi - 3iy - 5y\).
- Combine real and imaginary parts: \((3x + 5y) + (5x - 3y)i = -6 + 24i\).
- Set up equations:
- Real part: \(3x + 5y = -6\)
- Imaginary part: \(5x - 3y = 24\)
- Solve these equations simultaneously to find \(x\) and \(y\).

2. **For \(-3 + ix^2y\) and \(x^2 + y + 4i\) being conjugates**:
- Set up equations:
- Real part: \(-3 = x^2 + y\)
- Imaginary part: \(x^2y = -4\)
- Substitute \(y\) from the first equation into the second and solve for \(x\) and \(y\).