Simplify each of the following and express each as a rational number:
( i ) `(frac(3)(2))^(4) xx (frac(1)(5))^(2)`
( ii ) `(frac(-2)(3))^(5) xx (frac(-3)(7))^(3)`
( iii ) `(frac(-1)(2))^(5) xx 2^(3) xx (frac(3)(4))^(2)`
( iv ) `(frac(2)(3))^(2) xx (frac(-3)(5))^(3) xx (frac(7)(2))^(2)`
( v ) `{(frac(-3)(4))^(3)-(frac(-5)(2))^(3)} xx 4^(2)`

Answer

Let's simplify each of the given expressions step by step.

### (i) \((\frac{3}{2})^{4} \times (\frac{1}{5})^{2}\)

1. **Apply the exponent to both the numerator and denominator:**
\[ = \frac{3^{4}}{2^{4}} \times \frac{1^{2}}{5^{2}} \]

2. **Calculate the powers:**
\[ = \frac{81}{16} \times \frac{1}{25} \]

3. **Multiply the fractions:**
\[ = \frac{81 \times 1}{16 \times 25} = \frac{81}{400} \]

### (ii) \((\frac{-2}{3})^{5} \times (\frac{-3}{7})^{3}\)

1. **Apply the exponent to both the numerator and denominator:**
\[ = \frac{(-2)^{5}}{3^{5}} \times \frac{(-3)^{3}}{7^{3}} \]

2. **Calculate the powers:**
\[ = \frac{-32}{243} \times \frac{-27}{343} \]

3. **Multiply the fractions:**
\[ = \frac{-32 \times -27}{243 \times 343} = \frac{864}{83187} \]

### (iii) \((\frac{-1}{2})^{5} \times 2^{3} \times (\frac{3}{4})^{2}\)

1. **Apply the exponent to both the numerator and denominator:**
\[ = \frac{(-1)^{5}}{2^{5}} \times 2^{3} \times \frac{3^{2}}{4^{2}} \]

2. **Calculate the powers:**
\[ = \frac{-1}{32} \times 8 \times \frac{9}{16} \]

3. **Multiply the fractions:**
\[ = \frac{-1 \times 8 \times 9}{32 \times 16} = \frac{-72}{512} = \frac{-9}{64} \]

### (iv) \((\frac{2}{3})^{2} \times (\frac{-3}{5})^{3} \times (\frac{7}{2})^{2}\)

1. **Apply the exponent to both the numerator and denominator:**
\[ = \frac{2^{2}}{3^{2}} \times \frac{-3^{3}}{5^{3}} \times \frac{7^{2}}{2^{2}} \]

2. **Calculate the powers:**
\[ = \frac{4}{9} \times \frac{-27}{125} \times \frac{49}{4} \]

3. **Multiply the fractions:**
\[ = \frac{4 \times -27 \times 49}{9 \times 125 \times 4} = \frac{-27 \times 49}{9 \times 125} = \frac{-1323}{1125} \]

### (v) \((\frac{-3}{4})^{3} - (\frac{-5}{2})^{3} \times 4^{2}\)

1. **Calculate each term separately:**
\[ = \frac{-27}{64} - \left(\frac{-125}{8} \times 16\right) \]

2. **Calculate the second term:**
\[ = \frac{-27}{64} - \frac{-125 \times 2}{8} = \frac{-27}{64} + \frac{250}{64} \]

3. **Combine the fractions:**
\[ = \frac{-27 + 250}{64} = \frac{223}{64} \]

### Final Answers:
1. (i) \(\frac{81}{400}\)
2. (ii) \(\frac{864}{83187}\)
3. (iii) \(\frac{-9}{64}\)
4. (iv) \(\frac{-1323}{1125}\)
5. (v) \(\frac{223}{64}\)