If `2^(x)-2^(x-1)=4`, then `x^(x)` is equal to

Options

A. `1`
B. `27`
C. `256`
D. none of these

Answer

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

### Step 1: Rewrite the equation
We can rewrite \(2^{x-1}\) as \(\frac{2^x}{2}\):
\[ 2^x - \frac{2^x}{2} = 4 \]

### Step 2: Factor out \(2^x\)
Now, factor out \(2^x\) from the left side:
\[ 2^x \left(1 - \frac{1}{2}\right) = 4 \]
This simplifies to:
\[ 2^x \left(\frac{1}{2}\right) = 4 \]

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

### Step 4: Rewrite 8 as a power of 2
We know that \(8\) can be expressed as \(2^3\):
\[ 2^x = 2^3 \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ x = 3 \]

### Step 6: Calculate \(x^x\)
Now that we have \(x = 3\), we can calculate \(x^x\):
\[ x^x = 3^3 = 27 \]

### Final Answer
Thus, \(x^x\) is equal to \(27\).

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