Find m, if `(2/9)^(6)div (2/9)^(3)= (2/9)^((m-1)/(m+1))`

Answer

To solve the equation \((\frac{2}{9})^{6} \div (\frac{2}{9})^{3} = (\frac{2}{9})^{\frac{m-1}{m+1}}\), we can 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 rewrite the left-hand side:

\[ (\frac{2}{9})^{6} \div (\frac{2}{9})^{3} = (\frac{2}{9})^{6-3} \]

### Step 2: Simplify the exponent
Now, simplify the exponent:

\[ (\frac{2}{9})^{6-3} = (\frac{2}{9})^{3} \]

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

\[ 3 = \frac{m-1}{m+1} \]

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

\[ 3(m + 1) = m - 1 \]

### Step 5: Distribute on the left side
Distributing the 3 on the left side gives us:

\[ 3m + 3 = m - 1 \]

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

\[ 3m - m = -1 - 3 \]

### Step 7: Combine like terms
This simplifies to:

\[ 2m = -4 \]

### Step 8: Solve for \(m\)
Now, divide both sides by 2 to find \(m\):

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

Thus, the value of \(m\) is \(-2\).

---