The value of m such that `(2/9)^3xx(4/81)^(-6)=(2/9)^(2m-1)` is

Options

A. -6
B. 0
C. -4
D. 3

Answer

To solve the equation \((\frac{2}{9})^3 \times (\frac{4}{81})^{-6} = (\frac{2}{9})^{2m-1}\), we will follow these steps:

### Step 1: Simplify \((\frac{4}{81})^{-6}\)

First, we can rewrite \(\frac{4}{81}\) in terms of \(\frac{2}{9}\):
\[ \frac{4}{81} = \frac{2^2}{(3^4)} = \left(\frac{2}{9}\right)^{2} \text{ (since } 81 = 9^2 \text{)} \]
Thus,
\[ (\frac{4}{81})^{-6} = \left(\left(\frac{2}{9}\right)^{2}\right)^{-6} = \left(\frac{2}{9}\right)^{-12} \]

### Step 2: Substitute back into the equation

Now we can substitute this back into the original equation:
\[ (\frac{2}{9})^3 \times (\frac{2}{9})^{-12} = (\frac{2}{9})^{2m-1} \]

### Step 3: Combine the exponents on the left side

Using the property of exponents \(a^m \times a^n = a^{m+n}\):
\[ (\frac{2}{9})^{3 + (-12)} = (\frac{2}{9})^{2m-1} \]
This simplifies to:
\[ (\frac{2}{9})^{-9} = (\frac{2}{9})^{2m-1} \]

### Step 4: Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other:
\[ -9 = 2m - 1 \]

### Step 5: Solve for \(m\)

Now, we will solve for \(m\):
\[ 2m = -9 + 1 \]
\[ 2m = -8 \]
\[ m = -4 \]

### Final Answer

Thus, the value of \(m\) is:
\[ \boxed{-4} \]