Express each of the following as a rational number:
( i ) `(frac(2)(3))^(3)`
( ii ) `(frac(-8)(5))^(3)`
( iii ) `(frac(-13)(11))^(2)`
( iv ) `(frac(1)(6))^(3)`
( v ) `(frac(-1)(2))^(5)` ( vi ) `(frac(-3)(4))^(4)`
( vii ) `(frac(-4)(7))^(3)`
( viii ) `(-1)^(9)`

Answer

Let's solve each part of the question step by step.

### (i) \((\frac{2}{3})^{3}\)

1. **Apply the exponent rule**: \((\frac{a}{b})^{n} = \frac{a^{n}}{b^{n}}\)
\[ = \frac{2^{3}}{3^{3}} \]

2. **Calculate \(2^{3}\) and \(3^{3}\)**:
\[ 2^{3} = 2 \times 2 \times 2 = 8 \]
\[ 3^{3} = 3 \times 3 \times 3 = 27 \]

3. **Combine the results**:
\[ = \frac{8}{27} \]

### (ii) \((\frac{-8}{5})^{3}\)

1. **Apply the exponent rule**:
\[ = \frac{(-8)^{3}}{5^{3}} \]

2. **Calculate \((-8)^{3}\) and \(5^{3}\)**:
\[ (-8)^{3} = -8 \times -8 \times -8 = -512 \]
\[ 5^{3} = 5 \times 5 \times 5 = 125 \]

3. **Combine the results**:
\[ = \frac{-512}{125} \]

### (iii) \((\frac{-13}{11})^{2}\)

1. **Apply the exponent rule**:
\[ = \frac{(-13)^{2}}{11^{2}} \]

2. **Calculate \((-13)^{2}\) and \(11^{2}\)**:
\[ (-13)^{2} = 169 \]
\[ 11^{2} = 121 \]

3. **Combine the results**:
\[ = \frac{169}{121} \]

### (iv) \((\frac{1}{6})^{3}\)

1. **Apply the exponent rule**:
\[ = \frac{1^{3}}{6^{3}} \]

2. **Calculate \(1^{3}\) and \(6^{3}\)**:
\[ 1^{3} = 1 \]
\[ 6^{3} = 216 \]

3. **Combine the results**:
\[ = \frac{1}{216} \]

### (v) \((\frac{-1}{2})^{5}\)

1. **Apply the exponent rule**:
\[ = \frac{(-1)^{5}}{2^{5}} \]

2. **Calculate \((-1)^{5}\) and \(2^{5}\)**:
\[ (-1)^{5} = -1 \]
\[ 2^{5} = 32 \]

3. **Combine the results**:
\[ = \frac{-1}{32} \]

### (vi) \((\frac{-3}{4})^{4}\)

1. **Apply the exponent rule**:
\[ = \frac{(-3)^{4}}{4^{4}} \]

2. **Calculate \((-3)^{4}\) and \(4^{4}\)**:
\[ (-3)^{4} = 81 \]
\[ 4^{4} = 256 \]

3. **Combine the results**:
\[ = \frac{81}{256} \]

### (vii) \((\frac{-4}{7})^{3}\)

1. **Apply the exponent rule**:
\[ = \frac{(-4)^{3}}{7^{3}} \]

2. **Calculate \((-4)^{3}\) and \(7^{3}\)**:
\[ (-4)^{3} = -64 \]
\[ 7^{3} = 343 \]

3. **Combine the results**:
\[ = \frac{-64}{343} \]

### (viii) \((-1)^{9}\)

1. **Calculate \((-1)^{9}\)**:
\[ = -1 \]

2. **Express as a rational number**:
\[ = \frac{-1}{1} \]

### Summary of Answers:
1. \((\frac{2}{3})^{3} = \frac{8}{27}\)
2. \((\frac{-8}{5})^{3} = \frac{-512}{125}\)
3. \((\frac{-13}{11})^{2} = \frac{169}{121}\)
4. \((\frac{1}{6})^{3} = \frac{1}{216}\)
5. \((\frac{-1}{2})^{5} = \frac{-1}{32}\)
6. \((\frac{-3}{4})^{4} = \frac{81}{256}\)
7. \((\frac{-4}{7})^{3} = \frac{-64}{343}\)
8. \((-1)^{9} = \frac{-1}{1}\)