If `4^(x + 3) xx 2^(x - 3) - 128 = 0` then the value of x is :

Options

A. `1/3`
B. `2/3`
C. `3/3`
D. `4/3`

Answer

To solve the equation \( 4^{(x + 3)} \cdot 2^{(x - 3)} - 128 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
First, we can rewrite \( 4 \) as \( 2^2 \). Thus, we have:
\[ 4^{(x + 3)} = (2^2)^{(x + 3)} = 2^{2(x + 3)} = 2^{(2x + 6)} \]
Now substituting this into the equation gives:
\[ 2^{(2x + 6)} \cdot 2^{(x - 3)} - 128 = 0 \]

### Step 2: Combine the powers of 2
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can combine the terms:
\[ 2^{(2x + 6 + x - 3)} - 128 = 0 \]
This simplifies to:
\[ 2^{(3x + 3)} - 128 = 0 \]

### Step 3: Rewrite 128 as a power of 2
Next, we know that \( 128 = 2^7 \). So we can rewrite the equation as:
\[ 2^{(3x + 3)} - 2^7 = 0 \]

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

### Step 5: Solve for x
Now, we can solve for \( x \):
\[ 3x = 7 - 3 \]
\[ 3x = 4 \]
\[ x = \frac{4}{3} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{4}{3}} \]

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