In each of the following find `r+s, r- s, rs, (r )/(s)` if r denotes the first complex number and s denotes the second complex number
`3+7i, i`

Answer

To solve the problem, we need to find the values of \( r+s \), \( r-s \), \( rs \), and \( \frac{r}{s} \) where \( r = 3 + 7i \) and \( s = i \).

### Step-by-step Solution:

1. **Identify the Complex Numbers**:
- Let \( r = 3 + 7i \)
- Let \( s = i \)

2. **Calculate \( r + s \)**:
\[ r + s = (3 + 7i) + i = 3 + 7i + 1i = 3 + 8i \]

3. **Calculate \( r - s \)**:
\[ r - s = (3 + 7i) - i = 3 + 7i - 1i = 3 + 6i \]

4. **Calculate \( rs \)**:
\[ rs = (3 + 7i)(i) = 3i + 7i^2 \]
Since \( i^2 = -1 \):
\[ rs = 3i + 7(-1) = 3i - 7 = -7 + 3i \]

5. **Calculate \( \frac{r}{s} \)**:
\[ \frac{r}{s} = \frac{3 + 7i}{i} \]
To simplify, multiply the numerator and the denominator by \( i \):
\[ \frac{r}{s} = \frac{(3 + 7i)i}{i^2} = \frac{3i + 7i^2}{-1} = \frac{3i - 7}{-1} = -3i + 7 = 7 - 3i \]

### Final Results:
- \( r + s = 3 + 8i \)
- \( r - s = 3 + 6i \)
- \( rs = -7 + 3i \)
- \( \frac{r}{s} = 7 - 3i \)