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 will follow these steps:

### Step 1: Factor 135
First, we need to express 135 in terms of its prime factors:
\[ 135 = 5^1 \cdot 3^3 \]

### Step 2: Rewrite the equation
Now, we can rewrite the original equation using the factorization of 135:
\[ 5^{(x-2)} \cdot 3^{(2x-3)} = 5^1 \cdot 3^3 \]

### Step 3: Equate the bases
Since the bases on both sides of the equation are the same, we can equate the exponents for the base 5 and the base 3 separately.

**For base 5:**
\[ x - 2 = 1 \]

**For base 3:**
\[ 2x - 3 = 3 \]

### Step 4: Solve the equations
Now, we will solve each equation for \( x \).

**From the first equation:**
\[ x - 2 = 1 \implies x = 1 + 2 = 3 \]

**From the second equation:**
\[ 2x - 3 = 3 \implies 2x = 3 + 3 \implies 2x = 6 \implies x = \frac{6}{2} = 3 \]

### Step 5: Conclusion
Both equations give us the same value for \( x \):
\[ x = 3 \]

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

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