Find the value of `x`, if `5^(x-2)xx3^(2x-3)=135`

Explanation

To solve the equation \( 5^{(x-2)} \cdot 3^{(2x-3)} = 135 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 5^{(x-2)} \cdot 3^{(2x-3)} = 135 \]

### Step 2: Factor the right-hand side
Next, we can express 135 in terms of its prime factors. We know that:
\[ 135 = 3^3 \cdot 5^1 \]
Thus, we rewrite the equation as:
\[ 5^{(x-2)} \cdot 3^{(2x-3)} = 3^3 \cdot 5^1 \]

### Step 3: Separate the bases
Now we can equate the powers of the same bases on both sides. This gives us two separate equations:
1. For the base 5:
\[ x - 2 = 1 \]
2. For the base 3:
\[ 2x - 3 = 3 \]

### Step 4: Solve the first equation
From the first equation:
\[ x - 2 = 1 \implies x = 1 + 2 = 3 \]

### Step 5: Solve the second equation
From the second equation:
\[ 2x - 3 = 3 \implies 2x = 3 + 3 = 6 \implies x = \frac{6}{2} = 3 \]

### Conclusion
Both equations yield the same result:
\[ x = 3 \]

Thus, the value of \( x \) is \( 3 \).

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