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

Options

A. 3
B. 5
C. 6
D. 7

Answer

To solve the equation \(5^{(x-3)} \cdot 3^{(2x-8)} = 225\), we can follow these steps:

### Step 1: Rewrite 225 in terms of its prime factors
First, we need to express 225 in terms of its prime factors. We know that:
\[ 225 = 15^2 = (3 \cdot 5)^2 = 3^2 \cdot 5^2 \]
Thus, we can rewrite the equation as:
\[ 5^{(x-3)} \cdot 3^{(2x-8)} = 3^2 \cdot 5^2 \]

### Step 2: Equate the exponents of the same bases
Now, we can equate the exponents of the same bases on both sides of the equation.

For the base 5:
\[ x - 3 = 2 \quad \text{(1)} \]

For the base 3:
\[ 2x - 8 = 2 \quad \text{(2)} \]

### Step 3: Solve the first equation
From equation (1):
\[ x - 3 = 2 \]
Adding 3 to both sides gives:
\[ x = 5 \]

### Step 4: Solve the second equation
From equation (2):
\[ 2x - 8 = 2 \]
Adding 8 to both sides gives:
\[ 2x = 10 \]
Dividing both sides by 2 gives:
\[ x = 5 \]

### Step 5: Conclusion
Both equations give us the same value of \(x\). Therefore, the solution to the equation is:
\[ \boxed{5} \]