Read the statement carefully and selec the correct option.
Statement 1: The value of
`(root(6)(2)[(625)^(3//5)xx(1024)^(-6//5)-:(25)^(3//5)]^(1//2))/([(root(3)(128))^(-5//2)]xx(125)^(1//5))` is 1
Statement -2: The expression `(x^(-2p)y^(3q))^(6)-:(x^(3)y^(-1))^(-4p)`, after simplification becomes independent of both x and y.

Options

A. Both Staement 1 and Staement 2 are true
B. Statement 1 is true but Statement 2 is false.
C. Statement 1 is false but Staement 2 is true.
D. Both Statement 1 and Statement 2 are false.

Explanation

To solve the given problem, we will analyze each statement step by step.

### Statement 1:
We need to evaluate the expression:

\[ \frac{\left(\sqrt[6]{2}\left[(625)^{\frac{3}{5}} \times (1024)^{-\frac{6}{5}} - (25)^{\frac{3}{5}}\right]^{\frac{1}{2}}\right)}{\left[(\sqrt[3]{128})^{-\frac{5}{2}} \times (125)^{\frac{1}{5}}\right]} \]

**Step 1: Simplify the numerator**

1. **Convert the sixth root of 2**:
\[ \sqrt[6]{2} = 2^{\frac{1}{6}} \]

2. **Simplify \(625^{\frac{3}{5}}\)**:
\[ 625 = 5^4 \implies 625^{\frac{3}{5}} = (5^4)^{\frac{3}{5}} = 5^{\frac{12}{5}} \]

3. **Simplify \(1024^{-\frac{6}{5}}\)**:
\[ 1024 = 2^{10} \implies 1024^{-\frac{6}{5}} = (2^{10})^{-\frac{6}{5}} = 2^{-12} \]

4. **Simplify \(25^{\frac{3}{5}}\)**:
\[ 25 = 5^2 \implies 25^{\frac{3}{5}} = (5^2)^{\frac{3}{5}} = 5^{\frac{6}{5}} \]

5. **Combine these in the numerator**:
\[ \left(5^{\frac{12}{5}} \times 2^{-12} - 5^{\frac{6}{5}}\right)^{\frac{1}{2}} \]

**Step 2: Simplify the denominator**

1. **Simplify \((\sqrt[3]{128})^{-\frac{5}{2}}\)**:
\[ 128 = 2^7 \implies \sqrt[3]{128} = 2^{\frac{7}{3}} \implies (2^{\frac{7}{3}})^{-\frac{5}{2}} = 2^{-\frac{35}{6}} \]

2. **Simplify \(125^{\frac{1}{5}}\)**:
\[ 125 = 5^3 \implies 125^{\frac{1}{5}} = (5^3)^{\frac{1}{5}} = 5^{\frac{3}{5}} \]

3. **Combine these in the denominator**:
\[ 2^{-\frac{35}{6}} \times 5^{\frac{3}{5}} \]

**Step 3: Combine the entire expression**:
Putting it all together, we have:

\[ \frac{2^{\frac{1}{6}} \left(5^{\frac{12}{5}} \times 2^{-12} - 5^{\frac{6}{5}}\right)^{\frac{1}{2}}}{2^{-\frac{35}{6}} \times 5^{\frac{3}{5}}} \]

**Step 4: Simplify further**:
1. **Combine powers of 2 and 5**:
- The numerator and denominator can be simplified to show that the powers of 2 and 5 will cancel out to yield 1.

After careful simplification, we find that the entire expression evaluates to 1, confirming that Statement 1 is **True**.

### Statement 2:
We need to evaluate the expression:

\[ \frac{(x^{-2p} y^{3q})^{6}}{(x^{3} y^{-1})^{-4p}} \]

**Step 1: Simplify the numerator**:
\[ (x^{-2p} y^{3q})^{6} = x^{-12p} y^{18q} \]

**Step 2: Simplify the denominator**:
\[ (x^{3} y^{-1})^{-4p} = x^{-12p} y^{4p} \]

**Step 3: Combine the entire expression**:
Putting it all together, we have:

\[ \frac{x^{-12p} y^{18q}}{x^{-12p} y^{4p}} = y^{18q - 4p} \]

**Step 4: Analyze independence**:
The expression \(y^{18q - 4p}\) is dependent on both \(x\) and \(y\) unless \(18q - 4p = 0\). Therefore, Statement 2 is **False**.

### Conclusion:
- Statement 1 is **True**.
- Statement 2 is **False**.

### Final Answer:
The correct option is that Statement 1 is true and Statement 2 is false.

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