`(49)^(2)xx (7)^(8)div (343)^(3)=(7)^(?)`

Options

A. 3
B. 11
C. 7
D. 9

Explanation

To solve the equation \( (49)^2 \times (7)^8 \div (343)^3 = (7)^? \), we will simplify each term step by step.

### Step 1: Rewrite the numbers in terms of base 7
- We know that \( 49 = 7^2 \) and \( 343 = 7^3 \).
- Therefore, we can rewrite the equation as follows:
\[ (7^2)^2 \times (7^8) \div (7^3)^3 \]

### Step 2: Simplify the powers
- Using the power of a power property \( (a^m)^n = a^{m \cdot n} \):
\[ (7^2)^2 = 7^{2 \cdot 2} = 7^4 \]
- And for \( (7^3)^3 \):
\[ (7^3)^3 = 7^{3 \cdot 3} = 7^9 \]

### Step 3: Substitute back into the equation
- Now we substitute back into the equation:
\[ 7^4 \times 7^8 \div 7^9 \]

### Step 4: Combine the terms
- We can combine \( 7^4 \) and \( 7^8 \) using the property \( a^m \times a^n = a^{m+n} \):
\[ 7^4 \times 7^8 = 7^{4+8} = 7^{12} \]
- Now we have:
\[ 7^{12} \div 7^9 \]

### Step 5: Simplify the division
- Using the property \( a^m \div a^n = a^{m-n} \):
\[ 7^{12} \div 7^9 = 7^{12-9} = 7^3 \]

### Step 6: Set the equation equal to \( 7^? \)
- Now we have:
\[ 7^3 = 7^? \]
- Therefore, we can conclude that:
\[ ? = 3 \]

### Final Answer
Thus, the value of \( ? \) is \( 3 \).

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