If `x=3+2sqrt(2)`, then find :
`(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 \( \frac{1}{x} \), \( x + \frac{1}{x} \), \( x - \frac{1}{x} \), and \( x^2 - \frac{1}{x^2} \) given that \( x = 3 + 2\sqrt{2} \).

### Step 1: Find \( \frac{1}{x} \)
Given \( x = 3 + 2\sqrt{2} \), we need to find \( \frac{1}{x} \).

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \cdot \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]

Now, calculate the denominator:

\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]

Thus,

\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

### Step 2: Find \( x + \frac{1}{x} \)
Now, we calculate \( x + \frac{1}{x} \):

\[ x + \frac{1}{x} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 3 + 3 + 2\sqrt{2} - 2\sqrt{2} = 6 \]

### Step 3: Find \( x - \frac{1}{x} \)
Next, we calculate \( x - \frac{1}{x} \):

\[ x - \frac{1}{x} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) = 3 - 3 + 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2} \]

### Step 4: Find \( x^2 - \frac{1}{x^2} \)
We can use the identity \( x^2 - \frac{1}{x^2} = (x - \frac{1}{x})(x + \frac{1}{x}) \).

From Steps 2 and 3, we have:

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

Now, substitute these values into the identity:

\[ x^2 - \frac{1}{x^2} = (4\sqrt{2})(6) = 24\sqrt{2} \]

### Summary of Results
1. \( \frac{1}{x} = 3 - 2\sqrt{2} \)
2. \( x + \frac{1}{x} = 6 \)
3. \( x - \frac{1}{x} = 4\sqrt{2} \)
4. \( x^2 - \frac{1}{x^2} = 24\sqrt{2} \)