If `2^(x)-2^(x-1)=8`, then the value of `x^(3)` is :

Options

A. 46
B. 27
C. 64
D. 8

Answer

To solve the equation \( 2^x - 2^{x-1} = 8 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 2^x - 2^{x-1} = 8 \]
We can rewrite \( 2^{x-1} \) as \( \frac{2^x}{2} \):
\[ 2^x - \frac{2^x}{2} = 8 \]

### Step 2: Factor out \( 2^x \)
Now, we can factor out \( 2^x \) from the left side:
\[ 2^x \left(1 - \frac{1}{2}\right) = 8 \]
This simplifies to:
\[ 2^x \left(\frac{1}{2}\right) = 8 \]

### Step 3: Multiply both sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ 2^x = 8 \times 2 \]
This simplifies to:
\[ 2^x = 16 \]

### Step 4: Express 16 as a power of 2
We know that \( 16 \) can be expressed as \( 2^4 \):
\[ 2^x = 2^4 \]

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

### Step 6: Calculate \( x^3 \)
Now, we need to find \( x^3 \):
\[ x^3 = 4^3 = 64 \]

### Final Answer
Thus, the value of \( x^3 \) is:
\[ \boxed{64} \]

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