Study the following statements carefully and select the correct option.
P: The multiplicative inverse of `(2/3)^(-5)` is `((-2)/(3))^5`
Q : The value of `[1^@ +2^@ + 3^@] xx 6^(-1)` is `1/2`

Options

A. Both P and Q are true.
B. Both P and Q are false.
C. P is true but Q is false.
D. P is false but Q is true.

Answer

To solve the question, we need to evaluate the two statements P and Q separately.

### Statement P:
**P: The multiplicative inverse of \((\frac{2}{3})^{-5}\) is \((- \frac{2}{3})^{5}\)**

1. **Understanding Multiplicative Inverse**: The multiplicative inverse of a number \(x\) is defined as \(\frac{1}{x}\). For example, the multiplicative inverse of \(2\) is \(\frac{1}{2}\) because \(2 \times \frac{1}{2} = 1\).

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

3. **Finding the Multiplicative Inverse**:
\[ \text{Multiplicative Inverse of } (\frac{2}{3})^{-5} = \frac{1}{(\frac{2}{3})^{-5}} = \frac{2^5}{3^5} \]

4. **Comparing with \((- \frac{2}{3})^{5}\)**:
\[ (-\frac{2}{3})^{5} = -\frac{2^5}{3^5} \]

5. **Conclusion for Statement P**: The multiplicative inverse of \((\frac{2}{3})^{-5}\) is \(\frac{2^5}{3^5}\), not \((- \frac{2}{3})^{5}\). Therefore, **Statement P is false**.

### Statement Q:
**Q: The value of \([1^0 + 2^0 + 3^0] \times 6^{-1}\) is \(\frac{1}{2}\)**

1. **Calculating \(1^0\), \(2^0\), and \(3^0\)**:
\[ 1^0 = 1, \quad 2^0 = 1, \quad 3^0 = 1 \]

2. **Summing the Values**:
\[ 1^0 + 2^0 + 3^0 = 1 + 1 + 1 = 3 \]

3. **Calculating \(6^{-1}\)**:
\[ 6^{-1} = \frac{1}{6} \]

4. **Final Calculation**:
\[ [1^0 + 2^0 + 3^0] \times 6^{-1} = 3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

5. **Conclusion for Statement Q**: The value is indeed \(\frac{1}{2}\). Therefore, **Statement Q is true**.

### Final Conclusion:
- Statement P is **false**.
- Statement Q is **true**.

### Correct Option:
Since P is false and Q is true, the correct option is that **only Q is true**.