Simiplify :
`(16xx2^(n+1)-4xx2^(n))/(16xx2^(n+2)-2xx2^(n+1))`

Explanation

To simplify the expression `(16 * 2^(n+1) - 4 * 2^n) / (16 * 2^(n+2) - 2 * 2^(n+1))`, we will follow these steps:

### Step 1: Rewrite the expression using properties of exponents
We can rewrite the terms in the numerator and denominator using the property of exponents, which states that \( a^{m+n} = a^m \cdot a^n \).

**Numerator:**
- \( 16 * 2^{n+1} = 16 * 2^n * 2^1 = 16 * 2^n * 2 \)
- \( 4 * 2^n = 4 * 2^n \)

So, the numerator becomes:
\[ 16 * 2^n * 2 - 4 * 2^n = 2^n (16 * 2 - 4) \]

**Denominator:**
- \( 16 * 2^{n+2} = 16 * 2^n * 2^2 = 16 * 2^n * 4 \)
- \( 2 * 2^{n+1} = 2 * 2^n * 2 = 2^{n+1} * 2 = 2^n * 2^2 \)

So, the denominator becomes:
\[ 16 * 2^n * 4 - 2 * 2^n * 2 = 2^n (16 * 4 - 2 * 2) \]

### Step 2: Substitute back into the expression
Now substituting back into the expression, we have:
\[ \frac{2^n (16 * 2 - 4)}{2^n (16 * 4 - 4)} \]

### Step 3: Cancel out the common factor
Since \( 2^n \) is common in both the numerator and denominator, we can cancel it out:
\[ \frac{16 * 2 - 4}{16 * 4 - 4} \]

### Step 4: Simplify the terms
Now we simplify the terms:
- Numerator: \( 16 * 2 - 4 = 32 - 4 = 28 \)
- Denominator: \( 16 * 4 - 4 = 64 - 4 = 60 \)

So, we have:
\[ \frac{28}{60} \]

### Step 5: Reduce the fraction
Now we can reduce the fraction \( \frac{28}{60} \):
- The greatest common divisor (GCD) of 28 and 60 is 4.
- Dividing both the numerator and denominator by 4 gives:
\[ \frac{28 \div 4}{60 \div 4} = \frac{7}{15} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{7}{15} \]

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