Let's find the values of the expressions step by step, given [tex]\( x = 3 - 2\sqrt{2} \)[/tex].
1. Calculate [tex]\( x + \frac{1}{x} \)[/tex]:
```
x + \frac{1}{x} \approx 6
```
2. Calculate [tex]\( x - \frac{1}{x} \)[/tex]:
```
x - \frac{1}{x} \approx -5.656854249492387
```
3. Calculate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
```
x^2 + \frac{1}{x^2} \approx 34
```
4. Calculate [tex]\( x^2 - \frac{1}{x^2} \)[/tex]:
```
x^2 - \frac{1}{x^2} \approx -33.94112549695436
```
5. Calculate [tex]\( x^4 + \frac{1}{x^4} \)[/tex]:
```
x^4 + \frac{1}{x^4} \approx 1154
```
So, we have calculated the following:
- [tex]\( x + \frac{1}{x} \approx 6.000000000000007 \)[/tex]
- [tex]\( x - \frac{1}{x} \approx -5.656854249492387 \)[/tex]
- [tex]\( x^2 + \frac{1}{x^2} \approx 34.00000000000008 \)[/tex]
- [tex]\( x^2 - \frac{1}{x^2} \approx -33.94112549695436 \)[/tex]
- [tex]\( x^4 + \frac{1}{x^4} \approx 1154.0000000000052 \)[/tex]
These are the values for the given expressions when [tex]\( x = 3 - 2\sqrt{2} \)[/tex].
