Simplify :
(i) `(4)/(9)divide((-5)/(12))`
(ii) `-8divide((-5)/(16))`
(iii) `(-12)/(7)divide (-18)`
(iv) `((-1)/(10))divide((-8)/(5))`
(v) `((-16)/(35))divide ((-15)/(14))`
(vi) `((-65)/(14))divide((13)/(-7))`

Answer

Let's simplify each part step by step.

### (i) \(\frac{4}{9} \div \left(-\frac{5}{12}\right)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(-\frac{5}{12}\).
\[ \frac{4}{9} \times \left(-\frac{12}{5}\right) \]

2. **Multiply**: Multiply the numerators and the denominators.
\[ = \frac{4 \times (-12)}{9 \times 5} = \frac{-48}{45} \]

3. **Simplify**: Divide both the numerator and denominator by their greatest common divisor (GCD), which is 3.
\[ = \frac{-16}{15} \]

### (ii) \(-8 \div \left(-\frac{5}{16}\right)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(-\frac{5}{16}\).
\[ -8 \times \left(-\frac{16}{5}\right) \]

2. **Multiply**:
\[ = \frac{-8 \times -16}{5} = \frac{128}{5} \]

### (iii) \(\frac{-12}{7} \div (-18)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(-18\).
\[ \frac{-12}{7} \times \left(-\frac{1}{18}\right) \]

2. **Multiply**:
\[ = \frac{-12 \times -1}{7 \times 18} = \frac{12}{126} \]

3. **Simplify**: Divide both the numerator and denominator by their GCD, which is 6.
\[ = \frac{2}{21} \]

### (iv) \(\left(-\frac{1}{10}\right) \div \left(-\frac{8}{5}\right)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(-\frac{8}{5}\).
\[ \left(-\frac{1}{10}\right) \times \left(-\frac{5}{8}\right) \]

2. **Multiply**:
\[ = \frac{1 \times 5}{10 \times 8} = \frac{5}{80} \]

3. **Simplify**: Divide both the numerator and denominator by their GCD, which is 5.
\[ = \frac{1}{16} \]

### (v) \(\left(-\frac{16}{35}\right) \div \left(-\frac{15}{14}\right)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(-\frac{15}{14}\).
\[ \left(-\frac{16}{35}\right) \times \left(-\frac{14}{15}\right) \]

2. **Multiply**:
\[ = \frac{16 \times 14}{35 \times 15} = \frac{224}{525} \]

### (vi) \(\left(-\frac{65}{14}\right) \div \left(\frac{13}{-7}\right)\)

1. **Reciprocal**: Change the division to multiplication by taking the reciprocal of \(\frac{13}{-7}\).
\[ \left(-\frac{65}{14}\right) \times \left(-\frac{7}{13}\right) \]

2. **Multiply**:
\[ = \frac{65 \times 7}{14 \times 13} \]

3. **Simplify**:
\[ = \frac{455}{182} \]
Now, divide both the numerator and denominator by their GCD, which is 91.
\[ = \frac{5}{2} \]

### Summary of Answers:
1. \(-\frac{16}{15}\)
2. \(\frac{128}{5}\)
3. \(\frac{2}{21}\)
4. \(\frac{1}{16}\)
5. \(\frac{224}{525}\)
6. \(\frac{5}{2}\)