If `(5/7)^(4x).(7/5)^(3x-1)=(7/5)^(6)`, then find the value of x which satisfies the equation.

Options

A. -1
B. -7
C. 1
D. 7

Answer

To solve the equation \((\frac{5}{7})^{4x} \cdot (\frac{7}{5})^{(3x-1)} = (\frac{7}{5})^{6}\), we will follow these steps:

### Step 1: Rewrite the equation
We can rewrite the left-hand side of the equation by expressing \((\frac{7}{5})^{(3x-1)}\) in terms of \((\frac{5}{7})\):
\[ (\frac{5}{7})^{4x} \cdot (\frac{7}{5})^{(3x-1)} = (\frac{5}{7})^{4x} \cdot (\frac{5}{7})^{-(3x-1)} \]

### Step 2: Combine the exponents
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:
\[ (\frac{5}{7})^{4x - (3x - 1)} = (\frac{5}{7})^{4x - 3x + 1} = (\frac{5}{7})^{x + 1} \]

### Step 3: Rewrite the right-hand side
The right-hand side of the equation is already in the form of \((\frac{7}{5})^{6}\). We can rewrite it as:
\[ (\frac{7}{5})^{6} = (\frac{5}{7})^{-6} \]

### Step 4: Set the exponents equal
Now we have:
\[ (\frac{5}{7})^{x + 1} = (\frac{5}{7})^{-6} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x + 1 = -6 \]

### Step 5: Solve for \(x\)
To find \(x\), we subtract 1 from both sides:
\[ x = -6 - 1 = -7 \]

### Final Answer
Thus, the value of \(x\) that satisfies the equation is:
\[ \boxed{-7} \]