What will come in place of (?) question mark.
`(3)^(2) xx (3)^(6) xx (9)^(2) div (27)^(2) = (3)^(?)`

Options

A. 4
B. 6
C. 7
D. 5

Answer

To solve the equation \( (3^2) \times (3^6) \times (9^2) \div (27^2) = 3^{?} \), we will convert all the terms to the base of 3 and then simplify.

### Step-by-step Solution:

1. **Convert \(9^2\) to base 3:**
\[ 9 = 3^2 \implies 9^2 = (3^2)^2 = 3^{2 \times 2} = 3^4 \]

2. **Convert \(27^2\) to base 3:**
\[ 27 = 3^3 \implies 27^2 = (3^3)^2 = 3^{3 \times 2} = 3^6 \]

3. **Rewrite the equation using base 3:**
Now substituting back into the equation:
\[ (3^2) \times (3^6) \times (3^4) \div (3^6) = 3^{?} \]

4. **Combine the powers of 3 in the numerator:**
Using the property \( a^m \times a^n = a^{m+n} \):
\[ 3^2 \times 3^6 \times 3^4 = 3^{2 + 6 + 4} = 3^{12} \]

5. **Now divide by \(3^6\):**
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[ \frac{3^{12}}{3^6} = 3^{12 - 6} = 3^6 \]

6. **Set the equation equal to \(3^{?}\):**
\[ 3^6 = 3^{?} \]
Therefore, \( ? = 6 \).

### Final Answer:
The value that will come in place of the question mark is \(6\).

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