What is the value of x if `4^(x+1)=8^(x)`?

Options

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

Explanation

To solve the equation \(4^{(x+1)} = 8^{x}\), we will rewrite the bases in terms of powers of 2.

### Step 1: Rewrite the bases
We know that:
- \(4 = 2^2\)
- \(8 = 2^3\)

Thus, we can rewrite the equation as:
\[ (2^2)^{(x+1)} = (2^3)^{x} \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify both sides:
\[ 2^{2(x+1)} = 2^{3x} \]

### Step 3: Simplify the exponents
Now we can simplify the left side:
\[ 2^{2x + 2} = 2^{3x} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x + 2 = 3x \]

### Step 5: Solve for \(x\)
Now, we will isolate \(x\):
1. Subtract \(2x\) from both sides:
\[ 2 = 3x - 2x \]
This simplifies to:
\[ 2 = x \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{2} \]

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