Find the value of `m` so that `((2)/(9))^(3) xx ((2)/(9) )^(-6) = ((2)/(9) )^(2m-1)`.

Answer

To solve the equation \(\left(\frac{2}{9}\right)^{3} \times \left(\frac{2}{9}\right)^{-6} = \left(\frac{2}{9}\right)^{2m-1}\), we will follow these steps:

### Step 1: Apply the property of exponents
We know that when multiplying two powers with the same base, we can add the exponents. Therefore, we can rewrite the left-hand side of the equation as:

\[ \left(\frac{2}{9}\right)^{3 + (-6)} = \left(\frac{2}{9}\right)^{2m - 1} \]

### Step 2: Simplify the exponent
Now, simplify the exponent on the left-hand side:

\[ 3 + (-6) = 3 - 6 = -3 \]

So, we have:

\[ \left(\frac{2}{9}\right)^{-3} = \left(\frac{2}{9}\right)^{2m - 1} \]

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

\[ -3 = 2m - 1 \]

### Step 4: Solve for \(m\)
Now, we will solve for \(m\). First, add 1 to both sides:

\[ -3 + 1 = 2m \]

This simplifies to:

\[ -2 = 2m \]

Now, divide both sides by 2:

\[ m = \frac{-2}{2} = -1 \]

### Final Answer
Thus, the value of \(m\) is:

\[ \boxed{-1} \]

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