`7^( 8.9)div(343)^(1.7)xx(49)^(4.8)=7^(?)`

Options

A. 13.4
B. 12.8
C. 11.4
D. 9.6

Explanation

To solve the equation \( 7^{8.9} \div (343)^{1.7} \times (49)^{4.8} = 7^{?} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 7
First, we recognize that \( 343 \) and \( 49 \) can be expressed as powers of \( 7 \):
- \( 343 = 7^3 \)
- \( 49 = 7^2 \)

### Step 2: Substitute the powers into the equation
Now we can rewrite the equation:
\[ 7^{8.9} \div (7^3)^{1.7} \times (7^2)^{4.8} \]

### 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:
\[ (7^3)^{1.7} = 7^{3 \cdot 1.7} = 7^{5.1} \]
\[ (7^2)^{4.8} = 7^{2 \cdot 4.8} = 7^{9.6} \]

### Step 4: Rewrite the expression
Now substituting back into the equation gives us:
\[ 7^{8.9} \div 7^{5.1} \times 7^{9.6} \]

### Step 5: Apply the division and multiplication of exponents
Using the property \( a^m \div a^n = a^{m-n} \) and \( a^m \times a^n = a^{m+n} \), we can simplify:
\[ 7^{8.9 - 5.1 + 9.6} \]

### Step 6: Calculate the exponent
Now we calculate the exponent:
\[ 8.9 - 5.1 = 3.8 \]
\[ 3.8 + 9.6 = 13.4 \]

### Step 7: Write the final equation
Thus, we have:
\[ 7^{?} = 7^{13.4} \]

### Step 8: Conclude the value of ?
From this, we can conclude that:
\[ ? = 13.4 \]

### Final Answer
The value of \( ? \) is \( 13.4 \).

---