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If [tex]x = 3 - 2 \sqrt{2}[/tex], find the value of:

1. [tex]x + \frac{1}{x}[/tex]
2. [tex]x - \frac{1}{x}[/tex]
3. [tex]x^2 + \frac{1}{x^2}[/tex]
4. [tex]x^2 - \frac{1}{x^2}[/tex]
5. [tex]x^4 + \frac{1}{x^4}[/tex]

Sagot :

GinnyAnsweridn
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].
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