Assertion : `(- (3)/(5) )^(2) xx (- (3)/(5) )^(-7) = (- (3)/(5) )^(2-7) = ( - (3)/(5) )^(-5)`
Reason : `x^(n) xx y^(-n) = (xy)^(-n)`

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If assertion is false but reason is true.

Answer

To solve the assertion and reason provided, we need to analyze each part step by step.

### Step 1: Analyze the Assertion
The assertion states:
\[ (-\frac{3}{5})^{2} \times (-\frac{3}{5})^{-7} = (-\frac{3}{5})^{2 - 7} = (-\frac{3}{5})^{-5} \]

**Explanation:**
1. We have two terms with the same base, \(-\frac{3}{5}\).
2. According to the laws of exponents, when multiplying two powers with the same base, we add the exponents:
\[ a^m \times a^n = a^{m+n} \]
3. Here, the exponents are \(2\) and \(-7\):
\[ 2 + (-7) = 2 - 7 = -5 \]
4. Therefore, we can rewrite the left side:
\[ (-\frac{3}{5})^{2} \times (-\frac{3}{5})^{-7} = (-\frac{3}{5})^{-5} \]

### Conclusion for Assertion:
The assertion is correct.

### Step 2: Analyze the Reason
The reason states:
\[ x^{n} \times y^{-n} = (xy)^{-n} \]

**Explanation:**
1. This statement claims that the product of a power of \(x\) and a negative power of \(y\) equals the negative power of the product \(xy\).
2. However, this is not generally true unless \(x\) and \(y\) are equal. For example:
- Let \(x = 3\) and \(y = 2\), and let \(n = 2\):
\[ 3^{2} \times 2^{-2} \neq (3 \times 2)^{-2} \]
- Calculating the left side:
\[ 3^{2} = 9 \quad \text{and} \quad 2^{-2} = \frac{1}{4} \quad \Rightarrow \quad 9 \times \frac{1}{4} = \frac{9}{4} \]
- Calculating the right side:
\[ (3 \times 2)^{-2} = 6^{-2} = \frac{1}{36} \]
3. Since \(\frac{9}{4} \neq \frac{1}{36}\), the reason is false.

### Conclusion for Reason:
The reason is incorrect.

### Final Conclusion:
- The assertion is true, but the reason is false.

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