If `p^(3)-(1)/(p^(3))=140` then find the value of `p-(1)/(p)`

Options

A. 6
B. 5
C. 8
D. 10

Answer

To solve the equation \( p^3 - \frac{1}{p^3} = 140 \) and find the value of \( p - \frac{1}{p} \), we can use the identity that relates these two expressions.

### Step-by-Step Solution:

1. **Let \( x = p - \frac{1}{p} \)**.
We need to express \( p^3 - \frac{1}{p^3} \) in terms of \( x \).

2. **Use the identity**:
The identity states that:
\[ p^3 - \frac{1}{p^3} = \left( p - \frac{1}{p} \right) \left( p^2 + 1 + \frac{1}{p^2} \right) \]
We can also express \( p^2 + 1 + \frac{1}{p^2} \) in terms of \( x \):
\[ p^2 + \frac{1}{p^2} = \left( p - \frac{1}{p} \right)^2 + 2 = x^2 + 2 \]
Therefore,
\[ p^3 - \frac{1}{p^3} = x \left( x^2 + 2 + 1 \right) = x (x^2 + 3) \]

3. **Set up the equation**:
From the problem statement, we have:
\[ x (x^2 + 3) = 140 \]

4. **Rearrange the equation**:
This gives us the cubic equation:
\[ x^3 + 3x - 140 = 0 \]

5. **Solve for \( x \)**:
We can try some integer values for \( x \) to find a solution:
- For \( x = 5 \):
\[ 5^3 + 3 \cdot 5 = 125 + 15 = 140 \]
This satisfies the equation.

6. **Conclusion**:
Thus, the value of \( p - \frac{1}{p} \) is \( x = 5 \).

### Final Answer:
\[ p - \frac{1}{p} = 5 \]