`(6xx6xx6xx6xx6xx6)^(4)xx(6xx6xx6xx6)^(5) div (6xx6)^(7)=(36)^(?)`

Options

A. 15
B. 2
C. 16
D. 29

Explanation

To solve the equation \((6 \times 6 \times 6 \times 6 \times 6)^{4} \times (6 \times 6 \times 6 \times 6)^{5} \div (6 \times 6)^{7} = (36)^{?}\), we will follow these steps:

### Step 1: Rewrite the expressions using exponents
We can express the repeated multiplication of 6 as exponents:
- \(6 \times 6 = 6^2\)
- \(6 \times 6 \times 6 \times 6 = 6^4\)

Thus, we can rewrite the equation as:
\[ (6^4)^{4} \times (6^4)^{5} \div (6^2)^{7} = (36)^{?} \]

### Step 2: Simplify the left side using the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[ (6^4)^{4} = 6^{16} \quad \text{and} \quad (6^4)^{5} = 6^{20} \]
So, we have:
\[ 6^{16} \times 6^{20} \div (6^2)^{7} \]

### Step 3: Combine the powers of 6
Using the product of powers rule \(a^m \times a^n = a^{m+n}\):
\[ 6^{16 + 20} = 6^{36} \]

Now, we need to simplify the division:
\[ (6^2)^{7} = 6^{14} \]
So, we have:
\[ 6^{36} \div 6^{14} = 6^{36 - 14} = 6^{22} \]

### Step 4: Rewrite 6 in terms of 36
We know that \(36 = 6^2\). Therefore, we can express \(6^{22}\) in terms of \(36\):
\[ 6^{22} = (6^2)^{11} = 36^{11} \]

### Step 5: Set the equation equal to the right side
Now we have:
\[ 36^{11} = (36)^{?} \]
Thus, we can conclude that:
\[ ? = 11 \]

### Final Answer
The value of \(?\) is \(11\).

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