Find m, if `((3)/(7))^(9)div((3)/(7))^(5)=((3)/(7))^((3m+2)/(m-2))`

Options

A. `4`
B. `10`
C. `-4`
D. `15`

Answer

To solve the equation \(\left(\frac{3}{7}\right)^{9} \div \left(\frac{3}{7}\right)^{5} = \left(\frac{3}{7}\right)^{\frac{3m+2}{m-2}}\), we will follow these steps:

### Step 1: Apply the property of exponents
Using the property of exponents that states \(a^m \div a^n = a^{m-n}\), we can simplify the left-hand side:

\[ \left(\frac{3}{7}\right)^{9} \div \left(\frac{3}{7}\right)^{5} = \left(\frac{3}{7}\right)^{9-5} \]

### Step 2: Simplify the exponent
Now, calculate \(9 - 5\):

\[ 9 - 5 = 4 \]

Thus, the equation becomes:

\[ \left(\frac{3}{7}\right)^{4} = \left(\frac{3}{7}\right)^{\frac{3m+2}{m-2}} \]

### Step 3: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ 4 = \frac{3m + 2}{m - 2} \]

### Step 4: Cross-multiply
To eliminate the fraction, we cross-multiply:

\[ 4(m - 2) = 3m + 2 \]

### Step 5: Distribute on the left side
Distributing \(4\) gives us:

\[ 4m - 8 = 3m + 2 \]

### Step 6: Rearrange the equation
Now, we will rearrange the equation to isolate \(m\):

\[ 4m - 3m = 2 + 8 \]

### Step 7: Simplify
This simplifies to:

\[ m = 10 \]

### Final Answer
The value of \(m\) is:

\[ \boxed{10} \]

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