Simplify
(i) `((81)/(49))^(-(3)/(2))` (ii) `(14641)^(0.25)` (iii) ` ((32)/(243))^(-(4)/(5))` (iv) `((7776)/(243))^(-(3)/(5))`

Answer

Let's simplify each part step by step.

### Part (i): Simplify \(\left(\frac{81}{49}\right)^{-\frac{3}{2}}\)

1. **Rewrite the base**:
\[ 81 = 9^2 \quad \text{and} \quad 49 = 7^2 \]
So,
\[ \frac{81}{49} = \frac{9^2}{7^2} = \left(\frac{9}{7}\right)^2 \]

2. **Apply the exponent**:
\[ \left(\frac{9}{7}\right)^2^{-\frac{3}{2}} = \left(\frac{9}{7}\right)^{-3} = \left(\frac{7}{9}\right)^3 \]

3. **Calculate the final result**:
\[ \left(\frac{7}{9}\right)^3 = \frac{7^3}{9^3} = \frac{343}{729} \]

### Part (ii): Simplify \(14641^{0.25}\)

1. **Rewrite the exponent**:
\[ 0.25 = \frac{1}{4} \]
So,
\[ 14641^{0.25} = 14641^{\frac{1}{4}} \]

2. **Recognize the base**:
\[ 14641 = 11^4 \]

3. **Apply the exponent**:
\[ (11^4)^{\frac{1}{4}} = 11^{4 \cdot \frac{1}{4}} = 11^1 = 11 \]

### Part (iii): Simplify \(\left(\frac{32}{243}\right)^{-\frac{4}{5}}\)

1. **Rewrite the base**:
\[ 32 = 2^5 \quad \text{and} \quad 243 = 3^5 \]
So,
\[ \frac{32}{243} = \frac{2^5}{3^5} = \left(\frac{2}{3}\right)^5 \]

2. **Apply the exponent**:
\[ \left(\frac{2}{3}\right)^5^{-\frac{4}{5}} = \left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 \]

3. **Calculate the final result**:
\[ \left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16} \]

### Part (iv): Simplify \(\left(\frac{7776}{243}\right)^{-\frac{3}{5}}\)

1. **Rewrite the base**:
\[ 7776 = 6^5 \quad \text{and} \quad 243 = 3^5 \]
So,
\[ \frac{7776}{243} = \frac{6^5}{3^5} = \left(\frac{6}{3}\right)^5 = 2^5 \]

2. **Apply the exponent**:
\[ (2^5)^{-\frac{3}{5}} = 2^{-3} = \frac{1}{2^3} \]

3. **Calculate the final result**:
\[ \frac{1}{2^3} = \frac{1}{8} \]

### Final Results:
1. \(\left(\frac{81}{49}\right)^{-\frac{3}{2}} = \frac{343}{729}\)
2. \(14641^{0.25} = 11\)
3. \(\left(\frac{32}{243}\right)^{-\frac{4}{5}} = \frac{81}{16}\)
4. \(\left(\frac{7776}{243}\right)^{-\frac{3}{5}} = \frac{1}{8}\)