`(216)^(4)div(36)^(4)xx(6)^(5)=(6)^(?)`

Options

A. 13
B. 11
C. 10
D. 9

Explanation

To solve the equation \((216)^{4} \div (36)^{4} \times (6)^{5} = (6)^{?}\), we will simplify the left-hand side step by step.

### Step 1: Rewrite the numbers in terms of base 6
First, we express \(216\) and \(36\) in terms of base \(6\):
- \(216 = 6^3\)
- \(36 = 6^2\)

### Step 2: Substitute the values into the equation
Now we substitute these values into the equation:
\[ (6^3)^{4} \div (6^2)^{4} \times (6^{5}) \]

### Step 3: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (6^{3 \cdot 4}) \div (6^{2 \cdot 4}) \times (6^{5}) = (6^{12}) \div (6^{8}) \times (6^{5}) \]

### Step 4: Apply the division rule
Using the division rule \(a^m \div a^n = a^{m-n}\):
\[ 6^{12 - 8} \times 6^{5} = 6^{4} \times 6^{5} \]

### Step 5: Apply the multiplication rule
Using the multiplication rule \(a^m \times a^n = a^{m+n}\):
\[ 6^{4 + 5} = 6^{9} \]

### Step 6: Set the equation equal to \(6^{?}\)
Now we have:
\[ 6^{9} = 6^{?} \]
This implies that:
\[ ? = 9 \]

### Final Answer
Thus, the value of the question mark is \(9\).

---