Find the value of `9x` , if `5^(x-3).3^(2x-8)=225` .

Answer

To solve the equation \( 5^{(x-3)} \cdot 3^{(2x-8)} = 225 \), we will 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
Now we can equate the given expression to the prime factorization of 225:

\[ 5^{(x-3)} \cdot 3^{(2x-8)} = 3^2 \cdot 5^2 \]

### Step 3: Compare the exponents
Since the bases are the same, we can set the exponents equal to each other. This gives us two equations:

1. From the \(5\) terms:
\[ x - 3 = 2 \]

2. From the \(3\) terms:
\[ 2x - 8 = 2 \]

### Step 4: Solve the first equation
Let's solve the first equation:

\[ x - 3 = 2 \implies x = 2 + 3 = 5 \]

### Step 5: Solve the second equation
Now, let's solve the second equation:

\[ 2x - 8 = 2 \implies 2x = 2 + 8 \implies 2x = 10 \implies x = \frac{10}{2} = 5 \]

### Step 6: Conclusion
Both equations give us the same value for \(x\), which is \(5\).

### Step 7: Find \(9x\)
Now, we need to find \(9x\):

\[ 9x = 9 \cdot 5 = 45 \]

Thus, the final answer is:

\[ \boxed{45} \]

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