If `3^(x+8)=27^(2x+1)`, the value of x is

Options

A. `7`
B. 3
C. `-2`
D. 1

Answer

To solve the equation \(3^{x+8} = 27^{2x+1}\), we can follow these steps:

### Step 1: Rewrite 27 in terms of base 3
We know that \(27\) can be expressed as \(3^3\). Therefore, we can rewrite the equation as:
\[ 3^{x+8} = (3^3)^{2x+1} \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the right side:
\[ 3^{x+8} = 3^{3(2x+1)} \]
This simplifies to:
\[ 3^{x+8} = 3^{6x + 3} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ x + 8 = 6x + 3 \]

### Step 4: Solve for x
Now, we will solve for \(x\):
1. Subtract \(x\) from both sides:
\[ 8 = 5x + 3 \]
2. Subtract 3 from both sides:
\[ 5 = 5x \]
3. Divide both sides by 5:
\[ x = 1 \]

### Conclusion
The value of \(x\) is \(1\).

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