If ` 25 ^(x+1) = (125)/( 5^(x)) , ` find the value of x.

Answer

To solve the equation \( 25^{(x+1)} = \frac{125}{5^x} \), we can follow these steps:

### Step 1: Rewrite the bases
First, we can express 25 and 125 in terms of base 5:
- \( 25 = 5^2 \)
- \( 125 = 5^3 \)

So, we can rewrite the equation as:
\[ (5^2)^{(x+1)} = \frac{5^3}{5^x} \]

### Step 2: Simplify the left side
Using the power of a power property, we can simplify the left side:
\[ 5^{2(x+1)} = \frac{5^3}{5^x} \]

### Step 3: Simplify the right side
On the right side, we can simplify the fraction:
\[ \frac{5^3}{5^x} = 5^{3-x} \]

Now, our equation looks like:
\[ 5^{2(x+1)} = 5^{3-x} \]

### Step 4: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 2(x + 1) = 3 - x \]

### Step 5: Expand and simplify
Expanding the left side:
\[ 2x + 2 = 3 - x \]

### Step 6: Solve for x
Now, we can solve for \( x \):
1. Add \( x \) to both sides:
\[ 2x + x + 2 = 3 \]
This simplifies to:
\[ 3x + 2 = 3 \]

2. Subtract 2 from both sides:
\[ 3x = 1 \]

3. Divide by 3:
\[ x = \frac{1}{3} \]

### Final Answer
The value of \( x \) is \( \frac{1}{3} \).

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