If `2^(x + 3) . 4^(2x - 5) = 2^(3x + 7)` , then the value of x is :

Options

A. 3
B. 4
C. 6
D. 7

Answer

To solve the equation \(2^{(x + 3)} \cdot 4^{(2x - 5)} = 2^{(3x + 7)}\), we can follow these steps:

### Step 1: Rewrite the equation
We know that \(4\) can be expressed as \(2^2\). Therefore, we can rewrite \(4^{(2x - 5)}\) as \((2^2)^{(2x - 5)}\).

\[ 4^{(2x - 5)} = (2^2)^{(2x - 5)} = 2^{2(2x - 5)} = 2^{(4x - 10)} \]

Now, substituting this back into the original equation gives us:

\[ 2^{(x + 3)} \cdot 2^{(4x - 10)} = 2^{(3x + 7)} \]

### Step 2: Combine the exponents
Using the property of exponents that states \(a^m \cdot a^n = a^{(m+n)}\), we can combine the left-hand side:

\[ 2^{(x + 3 + 4x - 10)} = 2^{(3x + 7)} \]

This simplifies to:

\[ 2^{(5x - 7)} = 2^{(3x + 7)} \]

### Step 3: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[ 5x - 7 = 3x + 7 \]

### Step 4: Solve for \(x\)
Now, we will 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 both sides by \(2\):
\[ x = 7 \]

### Final Answer
The value of \(x\) is \(7\).

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