Simplify: `frac(16 xx 2^(n+1)-8 xx 2^(n))(16 xx 2^(n+2)-4 xx 2^(n+1))`

Options

A. `frac(4)(7)`
B. `frac(1)(7)`
C. `frac(2)(7)`
D. `frac(3)(7)`

Answer

To simplify the expression \(\frac{16 \cdot 2^{n+1} - 8 \cdot 2^n}{16 \cdot 2^{n+2} - 4 \cdot 2^{n+1}}\), we can follow these steps:

### Step 1: Factor out common terms in the numerator
In the numerator \(16 \cdot 2^{n+1} - 8 \cdot 2^n\), we can factor out \(8 \cdot 2^n\):
\[ 8 \cdot 2^n (2^{1} - 1) = 8 \cdot 2^n (2 - 1) = 8 \cdot 2^n \cdot 1 = 8 \cdot 2^n \]

### Step 2: Factor out common terms in the denominator
In the denominator \(16 \cdot 2^{n+2} - 4 \cdot 2^{n+1}\), we can factor out \(4 \cdot 2^{n+1}\):
\[ 4 \cdot 2^{n+1} (4 - 1) = 4 \cdot 2^{n+1} \cdot 3 = 12 \cdot 2^{n+1} \]

### Step 3: Rewrite the expression
Now we can rewrite the expression using the factored forms:
\[ \frac{8 \cdot 2^n}{12 \cdot 2^{n+1}} \]

### Step 4: Simplify the fraction
We can simplify the fraction:
\[ \frac{8}{12} \cdot \frac{2^n}{2^{n+1}} = \frac{2}{3} \cdot \frac{1}{2} = \frac{2}{6} = \frac{1}{3} \]

### Final Result
Thus, the simplified form of the expression is:
\[ \frac{1}{3} \]

---