If `9^(n)= 27^(n+1)`, then `2^(n)=`

Options

A. `-(10)/(3)`
B. `-(8)/(3)`
C. `-(3)/(8)`
D. `(1)/(8)`

Explanation

To solve the equation \( 9^n = 27^{n+1} \) and find the value of \( 2^n \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 3
We know that:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

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

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

### Step 3: Simplify the right side
Now, simplify the right side:
\[ 3^{2n} = 3^{3n + 3} \]

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

### Step 5: Solve for \( n \)
Rearranging the equation gives:
\[ 2n - 3n = 3 \\ -n = 3 \\ n = -3 \]

### Step 6: Find \( 2^n \)
Now that we have \( n = -3 \), we can find \( 2^n \):
\[ 2^n = 2^{-3} \]

### Step 7: Apply the negative exponent rule
Using the rule \( a^{-m} = \frac{1}{a^m} \):
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

### Final Answer
Thus, the value of \( 2^n \) is:
\[ \frac{1}{8} \]

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