Find `x` such that `(frac(3)(5))^(3) xx (frac(3)(5))^(-6)=(frac(3)(5))^(2x-1).`

Answer

To solve the equation \((\frac{3}{5})^{3} \times (\frac{3}{5})^{-6} = (\frac{3}{5})^{2x-1}\), we will follow these steps:

### Step 1: Apply the Rule of Exponents
We know that when multiplying powers with the same base, we can add the exponents. Thus, we can rewrite the left-hand side:

\[ (\frac{3}{5})^{3} \times (\frac{3}{5})^{-6} = (\frac{3}{5})^{3 + (-6)} = (\frac{3}{5})^{3 - 6} \]

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

\[ 3 - 6 = -3 \]

So, the left-hand side becomes:

\[ (\frac{3}{5})^{-3} \]

### Step 3: Rewrite the Equation
Now, we can rewrite the equation as:

\[ (\frac{3}{5})^{-3} = (\frac{3}{5})^{2x - 1} \]

### Step 4: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:

\[ -3 = 2x - 1 \]

### Step 5: Solve for \(x\)
Now, we will solve for \(x\). First, add 1 to both sides:

\[ -3 + 1 = 2x \]

This simplifies to:

\[ -2 = 2x \]

Now, divide both sides by 2:

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

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

\[ \boxed{-1} \]

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