`(1)/(11)` of `[(17424)^(1//2)div(66)^(2)xx3^(3)]=?^(2)`

Options

A. `1/11`
B. `3/11`
C. `2/11`
D. `4/11`

Explanation

To solve the given question step-by-step, we will follow the mathematical operations as described in the video transcript.

### Step 1: Understand the Expression
We start with the expression:
\[ \frac{1}{11} \left( \frac{(17424)^{1/2}}{(66)^2 \times 3^3} \right) = ?^2 \]
Let \( x = ? \). Therefore, we can rewrite the equation as:
\[ x^2 = \frac{1}{11} \left( \frac{(17424)^{1/2}}{(66)^2 \times 3^3} \right) \]

### Step 2: Calculate \( (17424)^{1/2} \)
We need to find the square root of 17424:
\[ (17424)^{1/2} = \sqrt{17424} \]
Calculating \( \sqrt{17424} \):
\[ \sqrt{17424} = 132 \]

### Step 3: Calculate \( (66)^2 \) and \( 3^3 \)
Next, we calculate \( (66)^2 \) and \( 3^3 \):
\[ (66)^2 = 4356 \]
\[ 3^3 = 27 \]

### Step 4: Substitute Values into the Expression
Now, we substitute these values back into our equation:
\[ x^2 = \frac{1}{11} \left( \frac{132}{4356 \times 27} \right) \]

### Step 5: Calculate the Denominator
Now we calculate \( 4356 \times 27 \):
\[ 4356 \times 27 = 117612 \]

### Step 6: Substitute and Simplify
Now we substitute back into the equation:
\[ x^2 = \frac{1}{11} \left( \frac{132}{117612} \right) \]

### Step 7: Simplify the Fraction
Now simplify \( \frac{132}{117612} \):
\[ \frac{132}{117612} = \frac{1}{891} \]
Thus, we have:
\[ x^2 = \frac{1}{11} \times \frac{1}{891} = \frac{1}{9801} \]

### Step 8: Solve for \( x \)
Taking the square root of both sides:
\[ x = \frac{1}{99} \]

### Final Answer
Thus, the value of \( ? \) is:
\[ ? = \frac{1}{99} \]

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