Assertion : `root(3) ((27)/( 216))` can be written as `((216)/(27))^(1//3)`.
Reason : `a^(1//n) = root(n) (a)`, where both represents `n^(th)` root 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 assertion and reason statement, we need to analyze both parts step by step.

### Step 1: Understand the Assertion
The assertion states that:
\[ \sqrt[3]{\frac{27}{216}} = \left(\frac{216}{27}\right)^{\frac{1}{3}} \]
We need to check if this equality holds true.

### Step 2: Simplify the Left Side
First, let's simplify the left side:
\[ \sqrt[3]{\frac{27}{216}} = \left(\frac{27}{216}\right)^{\frac{1}{3}} \]
Now, we can simplify \(\frac{27}{216}\):
\[ \frac{27}{216} = \frac{1}{8} \quad \text{(since } 27 = 3^3 \text{ and } 216 = 6^3 = (3 \cdot 2)^3 = 3^3 \cdot 2^3\text{)} \]
Thus,
\[ \sqrt[3]{\frac{27}{216}} = \left(\frac{1}{8}\right)^{\frac{1}{3}} = \frac{1}{2} \]

### Step 3: Simplify the Right Side
Now let's simplify the right side:
\[ \left(\frac{216}{27}\right)^{\frac{1}{3}} = \left(8\right)^{\frac{1}{3}} = 2 \]

### Step 4: Compare Both Sides
From our simplifications:
- Left Side: \(\frac{1}{2}\)
- Right Side: \(2\)

Since \(\frac{1}{2} \neq 2\), the assertion is false.

### Step 5: Analyze the Reason
The reason states:
\[ a^{\frac{1}{n}} = \sqrt[n]{a} \]
This statement is indeed true, as it defines the relationship between exponents and roots.

### Conclusion
- The assertion is **false**.
- The reason is **true**.

### Final Answer
Assertion: False
Reason: True

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