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

Options

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

Answer

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

### Step 1: Combine the left-hand side using the properties of exponents
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the left-hand side:

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

### Step 2: Set the exponents equal to each other
Now we have:

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

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 can solve for \(x\):

1. Add 2 to both sides:

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

2. Divide both sides by 5:

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

Thus, the value of \(x\) is \(\frac{3}{5}\).

### Final Answer:
\[ x = \frac{3}{5} \]

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