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

Options

A. 3
B. 13
C. 7
D. 9

Answer

To solve the equation \( (49)^2 \times (7)^8 \div (343)^3 = (7)^{?} \), we will follow these steps:

### Step 1: Rewrite the numbers in terms of base 7
First, we need to express all the numbers in the equation using base 7.

- \( 49 \) can be rewritten as \( 7^2 \).
- \( 343 \) can be rewritten as \( 7^3 \).

So, we can rewrite the expression as:
\[ (7^2)^2 \times (7^8) \div (7^3)^3 \]

### Step 2: Simplify the expression
Now, we simplify each part of the expression:

- \( (7^2)^2 = 7^{2 \times 2} = 7^4 \)
- \( (7^3)^3 = 7^{3 \times 3} = 7^9 \)

Now, substituting these back into the expression gives:
\[ 7^4 \times 7^8 \div 7^9 \]

### Step 3: Combine the powers
Next, we combine the powers in the numerator:
\[ 7^4 \times 7^8 = 7^{4 + 8} = 7^{12} \]

Now, we have:
\[ \frac{7^{12}}{7^9} \]

### Step 4: Apply the division rule for exponents
When dividing powers with the same base, we subtract the exponents:
\[ 7^{12 - 9} = 7^3 \]

### Step 5: Set the equation equal to \( 7^{?} \)
Now we can equate the expression to \( 7^{?} \):
\[ 7^3 = 7^{?} \]

### Step 6: Solve for the question mark
From the equation above, we can see that:
\[ ? = 3 \]

Thus, the value of the question mark is \( 3 \).

### Final Answer:
\[ ? = 3 \]

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