If `p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot]` denotes the greatest integer function, then the value of `p(1-f)` is equal to

Options

A. `1`
B. `2`
C. `2^n`
D. `2^(2n)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define p and f
We start with the expression given in the question:
\[ p = (8 + 3\sqrt{7})^n \]
We also have:
\[ f = p - [p] \]
where \([p]\) denotes the greatest integer function (GIF) of \(p\).

### Step 2: Identify the conjugate
Next, we consider the conjugate of \(8 + 3\sqrt{7}\):
\[ q = 8 - 3\sqrt{7} \]
Now, we will analyze the product of \(p\) and \(q\).

### Step 3: Calculate \(p \cdot q\)
We can compute:
\[ p \cdot q = (8 + 3\sqrt{7})^n \cdot (8 - 3\sqrt{7})^n \]
Using the difference of squares:
\[ p \cdot q = (8^2 - (3\sqrt{7})^2)^n \]
Calculating \(8^2\) and \((3\sqrt{7})^2\):
\[ 8^2 = 64 \]
\[ (3\sqrt{7})^2 = 9 \cdot 7 = 63 \]
Thus,
\[ p \cdot q = (64 - 63)^n = 1^n = 1 \]

### Step 4: Express \(p\) in terms of \(q\)
From the product \(p \cdot q = 1\), we can express \(q\) in terms of \(p\):
\[ q = \frac{1}{p} \]

### Step 5: Analyze \(f\)
Since \(p\) is a positive quantity and \(q\) is less than 1 (because \(8 - 3\sqrt{7} < 1\)), we know:
\[ 0 < q < 1 \]
This implies:
\[ 0 < f < 1 \]

### Step 6: Calculate \(p(1 - f)\)
Now we can compute:
\[ p(1 - f) = p \cdot (1 - (p - [p])) \]
This simplifies to:
\[ p(1 - f) = p \cdot \left(1 - p + [p]\right) = p \cdot \left([p] + 1 - p\right) \]

### Step 7: Substitute \(p\) and \(q\)
Since \(f = p - [p]\), we can write:
\[ p(1 - f) = p \cdot q \]
From our earlier calculation, we found:
\[ p \cdot q = 1 \]
Thus,
\[ p(1 - f) = 1 \]

### Conclusion
The value of \(p(1 - f)\) is equal to:
\[ \boxed{1} \]