If `x=sqrt(2)+1`, then find the values of the following :
`(i)(1)/(x)" "(ii)x+(1)/(x)" "(iii)x-(1)/(x)" "(iv)x^(2)+(1)/(x^(2))`

Explanation

To solve the problem step by step, we will find the values of the expressions given that \( x = \sqrt{2} + 1 \).

### Step 1: Find \( \frac{1}{x} \)

Given \( x = \sqrt{2} + 1 \), we can find \( \frac{1}{x} \) by rationalizing the denominator.

\[ \frac{1}{x} = \frac{1}{\sqrt{2} + 1} \]

To rationalize, multiply the numerator and denominator by \( \sqrt{2} - 1 \):

\[ \frac{1}{x} = \frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1 \]

### Step 2: Find \( x + \frac{1}{x} \)

Now, we can find \( x + \frac{1}{x} \):

\[ x + \frac{1}{x} = (\sqrt{2} + 1) + (\sqrt{2} - 1) \]

Combine like terms:

\[ x + \frac{1}{x} = \sqrt{2} + 1 + \sqrt{2} - 1 = 2\sqrt{2} \]

### Step 3: Find \( x - \frac{1}{x} \)

Next, we find \( x - \frac{1}{x} \):

\[ x - \frac{1}{x} = (\sqrt{2} + 1) - (\sqrt{2} - 1) \]

Combine like terms:

\[ x - \frac{1}{x} = \sqrt{2} + 1 - \sqrt{2} + 1 = 2 \]

### Step 4: Find \( x^2 + \frac{1}{x^2} \)

To find \( x^2 + \frac{1}{x^2} \), we can use the identity:

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

We already found \( x + \frac{1}{x} = 2\sqrt{2} \). Now square it:

\[ \left( x + \frac{1}{x} \right)^2 = (2\sqrt{2})^2 = 8 \]

Now, substitute into the identity:

\[ x^2 + \frac{1}{x^2} = 8 - 2 = 6 \]

### Summary of Results

1. \( \frac{1}{x} = \sqrt{2} - 1 \)
2. \( x + \frac{1}{x} = 2\sqrt{2} \)
3. \( x - \frac{1}{x} = 2 \)
4. \( x^2 + \frac{1}{x^2} = 6 \)