`(121)^(3)xx11 div (1331)^(2)=(11)^(?)`

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the equation \((121)^3 \times 11 \div (1331)^2 = (11)^{?}\), we will simplify the left-hand side step by step.

### Step 1: Rewrite the numbers in terms of powers of 11
First, we express \(121\) and \(1331\) as powers of \(11\):
- \(121 = 11^2\)
- \(1331 = 11^3\)

### Step 2: Substitute the powers into the equation
Now we can rewrite the equation:
\[ (11^2)^3 \times 11 \div (11^3)^2 \]

### Step 3: Simplify the powers
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify:
- \((11^2)^3 = 11^{2 \cdot 3} = 11^6\)
- \((11^3)^2 = 11^{3 \cdot 2} = 11^6\)

So, we can rewrite the equation as:
\[ 11^6 \times 11^1 \div 11^6 \]

### Step 4: Combine the powers in the numerator
Using the property \(a^m \times a^n = a^{m+n}\), we combine the powers in the numerator:
\[ 11^{6 + 1} \div 11^6 = 11^7 \div 11^6 \]

### Step 5: Simplify the division
Using the property \(a^m \div a^n = a^{m-n}\), we simplify:
\[ 11^{7 - 6} = 11^1 \]

### Step 6: Set the equation equal to \(11^{?}\)
Now we have:
\[ 11^1 = 11^{?} \]
This implies that \(? = 1\).

### Final Answer
Thus, the value of the question mark is:
\[ \boxed{1} \]