If `p ^(2) + (1)/(p ^(2))= 1 ( p gt 1)` then the value of `p ^(3) + (1)/(p ^(3))` is :

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the equation \( p^2 + \frac{1}{p^2} = 1 \) for \( p > 1 \) and find the value of \( p^3 + \frac{1}{p^3} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ p^2 + \frac{1}{p^2} = 1 \]

### Step 2: Use the identity for \( p + \frac{1}{p} \)
We know the identity:
\[ p^2 + \frac{1}{p^2} = \left( p + \frac{1}{p} \right)^2 - 2 \]
Let \( x = p + \frac{1}{p} \). Then we can rewrite the equation as:
\[ x^2 - 2 = 1 \]

### Step 3: Solve for \( x^2 \)
From the equation \( x^2 - 2 = 1 \), we can simplify it to:
\[ x^2 = 3 \]

### Step 4: Find \( x \)
Taking the square root of both sides, we have:
\[ x = \sqrt{3} \quad \text{(since \( p > 1 \), \( x \) must be positive)} \]

### Step 5: Use the identity for \( p^3 + \frac{1}{p^3} \)
We can use the identity:
\[ p^3 + \frac{1}{p^3} = \left( p + \frac{1}{p} \right) \left( p^2 + \frac{1}{p^2} \right) - \left( p + \frac{1}{p} \right) \]
Substituting \( x \) and \( p^2 + \frac{1}{p^2} \):
\[ p^3 + \frac{1}{p^3} = x \cdot 1 - x = x - x = 0 \]

### Final Result
Thus, the value of \( p^3 + \frac{1}{p^3} \) is:
\[ \boxed{0} \]