Assertion : `{ ((1)/(2))^6 }` is the reciprocal of `2^(36)`.
Reason : `{ (a^m)^n } = a^( m xx n)`, for any value of `a`.

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 problem, we need to analyze the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that \(\left(\frac{1}{2}\right)^6\) is the reciprocal of \(2^{36}\). For two numbers \(x\) and \(y\) to be reciprocals, their product must equal 1, i.e., \(x \cdot y = 1\).

### Step 2: Calculate the Product
We need to check if:
\[ \left(\frac{1}{2}\right)^6 \cdot 2^{36} = 1 \]

### Step 3: Rewrite the Expression
We can rewrite \(\left(\frac{1}{2}\right)^6\) as:
\[ \left(\frac{1}{2}\right)^6 = 2^{-6} \]
Thus, the product becomes:
\[ 2^{-6} \cdot 2^{36} \]

### Step 4: Apply the Laws of Exponents
Using the law of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:
\[ 2^{-6 + 36} = 2^{30} \]

### Step 5: Evaluate the Result
Since \(2^{30} \neq 1\), we conclude that:
\[ \left(\frac{1}{2}\right)^6 \cdot 2^{36} \neq 1 \]
Thus, the assertion is false.

### Step 6: Understand the Reason
The reason given is that \((a^m)^n = a^{m \cdot n}\) for any value of \(a\). This is a true statement based on the laws of exponents.

### Conclusion
- **Assertion**: False (because \(\left(\frac{1}{2}\right)^6\) is not the reciprocal of \(2^{36}\)).
- **Reason**: True (the exponent rule is correct).

### Final Answer
Assertion is FALSE, but Reason is TRUE.

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