If `p+(1)/(p)=112`, find `(p-112)^(15)+(1)/(p^(15))`.

Options

A. 1
B. 15
C. 10
D. 0

Answer

To solve the equation \( p + \frac{1}{p} = 112 \) and find the value of \( (p - 112)^{15} + \frac{1}{p^{15}} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ p + \frac{1}{p} = 112 \]

### Step 2: Isolate \( p - 112 \)
We can express \( p \) in terms of \( p - 112 \):
\[ p = 112 - \frac{1}{p} \]

### Step 3: Find \( p^{15} \)
To find \( p^{15} \), we can use the identity:
\[ p^2 + 1 = 112p \]
This can be rearranged to form a quadratic equation:
\[ p^2 - 112p + 1 = 0 \]
Using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ p = \frac{112 \pm \sqrt{112^2 - 4 \cdot 1}}{2} \]
Calculating the discriminant:
\[ 112^2 - 4 = 12544 - 4 = 12540 \]
Thus,
\[ p = \frac{112 \pm \sqrt{12540}}{2} \]

### Step 4: Calculate \( (p - 112)^{15} + \frac{1}{p^{15}} \)
Using the identity \( p + \frac{1}{p} = 112 \), we can derive:
\[ \frac{1}{p^{15}} = \frac{1}{(p + \frac{1}{p})^{15}} = 0 \]
This implies that:
\[ (p - 112)^{15} + \frac{1}{p^{15}} = 0 \]

### Final Answer
Thus, the value of \( (p - 112)^{15} + \frac{1}{p^{15}} \) is:
\[ \boxed{0} \]