If `2^(x+3) cdot 2^((2)(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 \( 2^{(x+3)} \cdot 2^{(2(2x - 5))} = 2^{(3x + 7)} \), we can follow these steps:

### Step 1: Simplify the Left Side
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents on the left side:

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

### Step 2: Expand the Exponent
Now, we need to simplify the exponent \( x + 3 + 2(2x - 5) \):

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

So, we have:

\[ x + 3 + 4x - 10 = 5x - 7 \]

Thus, the left side simplifies to:

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

### Step 3: Set the Exponents Equal
Now, we can set the exponents equal to each other since the bases are the same:

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

### Step 4: Solve for \( x \)
Now, we will isolate \( x \):

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
The value of \( x \) is \( 7 \).

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