`(3^(4) xx 9^(7)) div 27^(6) = 3^(?)`

Options

A. 2
B. 3
C. 0
D. 6

Answer

To solve the equation \((3^{4} \times 9^{7}) \div 27^{6} = 3^{?}\), we will follow these steps:

### Step 1: Rewrite the bases in terms of 3
First, we need to express \(9\) and \(27\) in terms of base \(3\):
- \(9 = 3^2\)
- \(27 = 3^3\)

### Step 2: Substitute the values into the equation
Now, we can rewrite the equation:
\[ (3^{4} \times (3^2)^{7}) \div (3^3)^{6} \]

### Step 3: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (3^{4} \times 3^{14}) \div 3^{18} \]
Here, \( (3^2)^{7} = 3^{2 \times 7} = 3^{14} \) and \( (3^3)^{6} = 3^{3 \times 6} = 3^{18} \).

### Step 4: Combine the powers in the numerator
Now, we can add the exponents in the numerator:
\[ 3^{4 + 14} \div 3^{18} \]
This simplifies to:
\[ 3^{18} \div 3^{18} \]

### Step 5: Apply the division property of exponents
Using the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[ 3^{18 - 18} = 3^{0} \]

### Step 6: Set the equation equal to \(3^{?}\)
Now we have:
\[ 3^{0} = 3^{?} \]
This implies:
\[ ? = 0 \]

### Final Answer
Thus, the value of the question mark is:
\[ \boxed{0} \]

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