Solve : `3^(x-1)xx5^(2y-3)=225.`

Answer

To solve the equation \( 3^{(x-1)} \times 5^{(2y-3)} = 225 \), we can follow these steps:

### Step 1: Rewrite 225 in terms of its prime factors
First, we need to express 225 as a product of its prime factors.

\[ 225 = 15 \times 15 = (3 \times 5) \times (3 \times 5) = 3^2 \times 5^2 \]

### Step 2: Set up the equation with the same bases
Now we can rewrite the original equation with the prime factorization of 225:

\[ 3^{(x-1)} \times 5^{(2y-3)} = 3^2 \times 5^2 \]

### Step 3: Compare the powers of the same bases
Since the bases on both sides of the equation are the same, we can equate the exponents for each base.

1. For the base 3:
\[ x - 1 = 2 \]

2. For the base 5:
\[ 2y - 3 = 2 \]

### Step 4: Solve for \( x \)
From the equation \( x - 1 = 2 \):

\[ x = 2 + 1 = 3 \]

### Step 5: Solve for \( y \)
From the equation \( 2y - 3 = 2 \):

\[ 2y = 2 + 3 = 5 \]
\[ y = \frac{5}{2} \]

### Final Solution
Thus, the values we found are:
\[ x = 3 \quad \text{and} \quad y = \frac{5}{2} \]

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