Write each of the following as a rational number with positive denominator.
(i) `frac(12)(-17)`
(ii) `frac(1)(-2)`
(iii) `frac(-8)(-19)`
(iv) `frac(11)(-6)`

Answer

To convert each of the given rational numbers to a form with a positive denominator, we will follow these steps for each fraction:

### Step-by-Step Solution:

**(i) Convert `frac(12)(-17)`**

1. The given fraction is \( \frac{12}{-17} \).
2. To make the denominator positive, we multiply both the numerator and the denominator by -1:
\[ \frac{12 \times (-1)}{-17 \times (-1)} = \frac{-12}{17} \]
3. Thus, \( \frac{12}{-17} = \frac{-12}{17} \).

**(ii) Convert `frac(1)(-2)`**

1. The given fraction is \( \frac{1}{-2} \).
2. Again, we multiply both the numerator and the denominator by -1:
\[ \frac{1 \times (-1)}{-2 \times (-1)} = \frac{-1}{2} \]
3. Thus, \( \frac{1}{-2} = \frac{-1}{2} \).

**(iii) Convert `frac(-8)(-19)`**

1. The given fraction is \( \frac{-8}{-19} \).
2. Here, both the numerator and denominator are negative. When we multiply both by -1:
\[ \frac{-8 \times (-1)}{-19 \times (-1)} = \frac{8}{19} \]
3. Thus, \( \frac{-8}{-19} = \frac{8}{19} \).

**(iv) Convert `frac(11)(-6)`**

1. The given fraction is \( \frac{11}{-6} \).
2. We multiply both the numerator and the denominator by -1:
\[ \frac{11 \times (-1)}{-6 \times (-1)} = \frac{-11}{6} \]
3. Thus, \( \frac{11}{-6} = \frac{-11}{6} \).

### Final Answers:
- (i) \( \frac{12}{-17} = \frac{-12}{17} \)
- (ii) \( \frac{1}{-2} = \frac{-1}{2} \)
- (iii) \( \frac{-8}{-19} = \frac{8}{19} \)
- (iv) \( \frac{11}{-6} = \frac{-11}{6} \)