Which is a solution to `(8^x)(2^4)=(1/2)^x` ?

Options

A. `-2`
B. `-1`
C. `-1/12`
D. 0

Answer

To solve the equation \( (8^x)(2^4) = \left( \frac{1}{2} \right)^x \), we can follow these steps:

### Step 1: Rewrite \( 8 \) as a power of \( 2 \)
We know that \( 8 = 2^3 \). Therefore, we can rewrite \( 8^x \) as:
\[ 8^x = (2^3)^x = 2^{3x} \]
Now, substitute this back into the equation:
\[ (2^{3x})(2^4) = \left( \frac{1}{2} \right)^x \]

### Step 2: Simplify the left-hand side
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can combine the terms on the left:
\[ 2^{3x + 4} = \left( \frac{1}{2} \right)^x \]

### Step 3: Rewrite the right-hand side
The expression \( \frac{1}{2} \) can be rewritten as \( 2^{-1} \). Thus, we have:
\[ \left( \frac{1}{2} \right)^x = (2^{-1})^x = 2^{-x} \]
Now, we can rewrite the equation as:
\[ 2^{3x + 4} = 2^{-x} \]

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

### Step 5: Solve for \( x \)
Now, we will solve for \( x \):
1. Add \( x \) to both sides:
\[ 3x + x + 4 = 0 \]
This simplifies to:
\[ 4x + 4 = 0 \]
2. Subtract \( 4 \) from both sides:
\[ 4x = -4 \]
3. Divide by \( 4 \):
\[ x = -1 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{-1} \]