If `3^(x+3) xx9^(2x-5) = 3^(3x+7 )`then the value of x is :

Options

A. 5
B. 6
C. 7
D. 4

Answer

To solve the equation \( 3^{(x+3)} \times 9^{(2x-5)} = 3^{(3x+7)} \), we can follow these steps:

### Step 1: Rewrite \( 9 \) in terms of base \( 3 \)
Since \( 9 = 3^2 \), we can rewrite \( 9^{(2x-5)} \) as:
\[ 9^{(2x-5)} = (3^2)^{(2x-5)} = 3^{2(2x-5)} = 3^{(4x-10)} \]

### Step 2: Substitute back into the equation
Now, we can substitute this back into the original equation:
\[ 3^{(x+3)} \times 3^{(4x-10)} = 3^{(3x+7)} \]

### Step 3: Combine the exponents on the left side
Using the property of exponents \( a^m \times a^n = a^{m+n} \), we can combine the left side:
\[ 3^{(x+3) + (4x-10)} = 3^{(3x+7)} \]
This simplifies to:
\[ 3^{(5x - 7)} = 3^{(3x + 7)} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 5x - 7 = 3x + 7 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
1. Subtract \( 3x \) from both sides:
\[ 5x - 3x - 7 = 7 \]
This simplifies to:
\[ 2x - 7 = 7 \]
2. Add \( 7 \) to both sides:
\[ 2x = 14 \]
3. Divide by \( 2 \):
\[ x = 7 \]

### Final Answer
Thus, the value of \( x \) is \( 7 \).

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