If `x = p + (1)/(p) and y = p - (1)/( p)` then the value of ` x^(4) - 2x^(2) y^(2) + y^(4)` is

Options

A. 24
B. 4
C. 16
D. 8

Answer

To solve the expression \( x^4 - 2x^2y^2 + y^4 \) given \( x = p + \frac{1}{p} \) and \( y = p - \frac{1}{p} \), we can follow these steps:

### Step 1: Calculate \( x^2 \) and \( y^2 \)

First, we will find \( x^2 \) and \( y^2 \).

\[ x^2 = \left( p + \frac{1}{p} \right)^2 = p^2 + 2 + \frac{1}{p^2} \]

\[ y^2 = \left( p - \frac{1}{p} \right)^2 = p^2 - 2 + \frac{1}{p^2} \]

### Step 2: Substitute \( x^2 \) and \( y^2 \) into the expression

Now we substitute \( x^2 \) and \( y^2 \) into the expression \( x^4 - 2x^2y^2 + y^4 \).

We know that:

\[ x^4 - 2x^2y^2 + y^4 = (x^2 - y^2)^2 \]

### Step 3: Calculate \( x^2 - y^2 \)

Now, we calculate \( x^2 - y^2 \):

\[ x^2 - y^2 = \left( p^2 + 2 + \frac{1}{p^2} \right) - \left( p^2 - 2 + \frac{1}{p^2} \right) \]
\[ = 2 + 2 = 4 \]

### Step 4: Calculate \( (x^2 - y^2)^2 \)

Now we can find \( (x^2 - y^2)^2 \):

\[ (x^2 - y^2)^2 = 4^2 = 16 \]

### Final Result

Thus, the value of \( x^4 - 2x^2y^2 + y^4 \) is:

\[ \boxed{16} \]

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