If `2^(x-2 ) = 5^(2-x)`, then find the value of `x`.

Answer

To solve the equation \( 2^{(x-2)} = 5^{(2-x)} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2^{(x-2)} = 5^{(2-x)} \]

### Step 2: Express both sides in terms of fractions
We can rewrite the equation as:
\[ \frac{2^x}{2^2} = \frac{5^2}{5^x} \]
This simplifies to:
\[ \frac{2^x}{4} = \frac{25}{5^x} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ 2^x \cdot 5^x = 25 \cdot 4 \]
This simplifies to:
\[ 2^x \cdot 5^x = 100 \]

### Step 4: Combine the bases
We can combine the bases on the left side:
\[ (2 \cdot 5)^x = 100 \]
This simplifies to:
\[ 10^x = 100 \]

### Step 5: Rewrite 100 as a power of 10
We know that:
\[ 100 = 10^2 \]
So we can rewrite the equation as:
\[ 10^x = 10^2 \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ x = 2 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{2} \]

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