What should come in place of the question mark(?) inthe following questions?
`(6)^(4)-:(36)^3 xx 216 = 6 ^(" ?-8")`

Options

A. 3
B. 6
C. 1
D. 4

Answer

To solve the equation \( (6^4) \div (36^3) \times 216 = 6^{(?-8)} \), we will follow these steps:

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

### Step 2: Substitute the values
Now we can rewrite the equation:
\[ (6^4) \div (6^2)^3 \times (6^3) = 6^{(?-8)} \]

### Step 3: Simplify the expression
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we simplify \( (6^2)^3 \):
\[ (6^4) \div (6^{2 \cdot 3}) \times (6^3) = 6^{(?-8)} \]
This simplifies to:
\[ (6^4) \div (6^6) \times (6^3) = 6^{(?-8)} \]

### Step 4: Apply the division property
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[ 6^{4-6} \times 6^3 = 6^{(?-8)} \]
This simplifies to:
\[ 6^{-2} \times 6^3 = 6^{(?-8)} \]

### Step 5: Combine the exponents
Using the property \( a^m \times a^n = a^{m+n} \):
\[ 6^{-2 + 3} = 6^{(?-8)} \]
This simplifies to:
\[ 6^{1} = 6^{(?-8)} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 1 = ? - 8 \]

### Step 7: Solve for the question mark
To find \( ? \), we add \( 8 \) to both sides:
\[ ? = 1 + 8 \]
Thus,
\[ ? = 9 \]

### Final Answer
The value that should come in place of the question mark is \( 9 \).

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