The solution of `(25)^(x -2) = (125)^(2x - 4)` :

Options

A. `3//4`
B. `0`
C. `2`
D. `-2`

Answer

To solve the equation \( (25)^{(x - 2)} = (125)^{(2x - 4)} \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 5
We know that:
- \( 25 = 5^2 \)
- \( 125 = 5^3 \)

So, we can rewrite the equation as:
\[ (5^2)^{(x - 2)} = (5^3)^{(2x - 4)} \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 5^{2(x - 2)} = 5^{3(2x - 4)} \]

This simplifies to:
\[ 5^{(2x - 4)} = 5^{(6x - 12)} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 2(x - 2) = 3(2x - 4) \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ 2x - 4 = 6x - 12 \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation to isolate \( x \):
\[ 2x - 6x = -12 + 4 \]
\[ -4x = -8 \]

### Step 6: Solve for \( x \)
Dividing both sides by -4 gives:
\[ x = 2 \]

### Conclusion
The solution to the equation \( (25)^{(x - 2)} = (125)^{(2x - 4)} \) is:
\[ \boxed{2} \]

---