If `x+((1)/(x)) =p`, then `x^(6) +((1)/(x^(6)))` equals

Options

A. `p^(6)+6p`
B. `p^(6)-6p`
C. `p^(6)+6p^(4) +9p^(2)+2`
D. `p^(6)-6p^(4)+9p^(2)-2`

Answer

To solve the problem, we need to find the value of \( x^6 + \frac{1}{x^6} \) given that \( x + \frac{1}{x} = p \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = p \]

2. **Use the identity for \( x^2 + \frac{1}{x^2} \):**
The formula states:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting \( p \) into the formula:
\[ x^2 + \frac{1}{x^2} = p^2 - 2 \]

3. **Next, use the identity for \( x^3 + \frac{1}{x^3} \):**
The formula states:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the values we have:
\[ x^3 + \frac{1}{x^3} = p \cdot (p^2 - 2) - p \]
Simplifying this:
\[ x^3 + \frac{1}{x^3} = p^3 - 2p - p = p^3 - 3p \]

4. **Now, use the identity for \( x^6 + \frac{1}{x^6} \):**
The formula states:
\[ x^6 + \frac{1}{x^6} = \left( x^3 + \frac{1}{x^3} \right)^2 - 2 \]
Substituting the value we found:
\[ x^6 + \frac{1}{x^6} = (p^3 - 3p)^2 - 2 \]

5. **Expand and simplify:**
\[ (p^3 - 3p)^2 = p^6 - 6p^4 + 9p^2 \]
Therefore,
\[ x^6 + \frac{1}{x^6} = p^6 - 6p^4 + 9p^2 - 2 \]

### Final Answer:
\[ x^6 + \frac{1}{x^6} = p^6 - 6p^4 + 9p^2 - 2 \]