Evaluate:
`(i) 5 ^(-3) " " (ii) ((1)/(3))^(-4) " "(iii) ((5)/(2))^(-3)" "(iv)(-2)^(-5)" "(v) ((-3)/(4)) ^(-4)`

Answer

Let's evaluate the given expressions step by step.

### (i) Evaluate \( 5^{-3} \)

1. **Apply the property of negative exponents**:
\[ 5^{-3} = \frac{1}{5^{3}} \]
2. **Calculate \( 5^3 \)**:
\[ 5^3 = 5 \times 5 \times 5 = 125 \]
3. **Final result**:
\[ 5^{-3} = \frac{1}{125} \]

### (ii) Evaluate \( \left(\frac{1}{3}\right)^{-4} \)

1. **Apply the property of negative exponents**:
\[ \left(\frac{1}{3}\right)^{-4} = \frac{1}{\left(\frac{1}{3}\right)^{4}} = 3^{4} \]
2. **Calculate \( 3^4 \)**:
\[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \]
3. **Final result**:
\[ \left(\frac{1}{3}\right)^{-4} = 81 \]

### (iii) Evaluate \( \left(\frac{5}{2}\right)^{-3} \)

1. **Apply the property of negative exponents**:
\[ \left(\frac{5}{2}\right)^{-3} = \frac{1}{\left(\frac{5}{2}\right)^{3}} = \left(\frac{2}{5}\right)^{3} \]
2. **Calculate \( \left(\frac{2}{5}\right)^{3} \)**:
\[ \left(\frac{2}{5}\right)^{3} = \frac{2^3}{5^3} = \frac{8}{125} \]
3. **Final result**:
\[ \left(\frac{5}{2}\right)^{-3} = \frac{8}{125} \]

### (iv) Evaluate \( (-2)^{-5} \)

1. **Apply the property of negative exponents**:
\[ (-2)^{-5} = \frac{1}{(-2)^{5}} \]
2. **Calculate \( (-2)^{5} \)**:
\[ (-2)^{5} = -2 \times -2 \times -2 \times -2 \times -2 = -32 \]
3. **Final result**:
\[ (-2)^{-5} = \frac{1}{-32} = -\frac{1}{32} \]

### (v) Evaluate \( \left(-\frac{3}{4}\right)^{-4} \)

1. **Apply the property of negative exponents**:
\[ \left(-\frac{3}{4}\right)^{-4} = \frac{1}{\left(-\frac{3}{4}\right)^{4}} = \left(-\frac{4}{3}\right)^{4} \]
2. **Calculate \( \left(-\frac{4}{3}\right)^{4} \)**:
\[ \left(-\frac{4}{3}\right)^{4} = \frac{(-4)^{4}}{3^{4}} = \frac{256}{81} \]
3. **Final result**:
\[ \left(-\frac{3}{4}\right)^{-4} = \frac{256}{81} \]

### Summary of Results:
1. \( 5^{-3} = \frac{1}{125} \)
2. \( \left(\frac{1}{3}\right)^{-4} = 81 \)
3. \( \left(\frac{5}{2}\right)^{-3} = \frac{8}{125} \)
4. \( (-2)^{-5} = -\frac{1}{32} \)
5. \( \left(-\frac{3}{4}\right)^{-4} = \frac{256}{81} \)