If `((5)/(7))^(4) xx ((5)/(7))^(-3) = ((5)/(7))^(5x -2)`, then x is equal to

Options

A. `(2)/(5)`
B. `(3)/(5)`
C. `(4)/(5)`
D. `(1)/(5)`

Answer

To solve the equation \(\left(\frac{5}{7}\right)^{4} \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{5x - 2}\), we can follow these steps:

### Step 1: Apply the Law of Exponents
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine the left side of the equation:

\[ \left(\frac{5}{7}\right)^{4} \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{4 + (-3)} = \left(\frac{5}{7}\right)^{4 - 3} = \left(\frac{5}{7}\right)^{1} \]

### Step 2: Set the Exponents Equal
Now we have:

\[ \left(\frac{5}{7}\right)^{1} = \left(\frac{5}{7}\right)^{5x - 2} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 1 = 5x - 2 \]

### Step 3: Solve for \(x\)
Now, we will solve for \(x\):

1. Add 2 to both sides:

\[ 1 + 2 = 5x \]
\[ 3 = 5x \]

2. Divide both sides by 5:

\[ x = \frac{3}{5} \]

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

\[ \boxed{\frac{3}{5}} \]