If ` x = 1 - sqrt(2) ` ,the value of ` ( x - (1)/( x))^(3)`

Options

A. `-8`
B. 8
C. `2 sqrt(2)`
D. 1

Answer

To solve the problem, we need to find the value of \( (x - \frac{1}{x})^3 \) given that \( x = 1 - \sqrt{2} \).

### Step-by-Step Solution:

1. **Find \( \frac{1}{x} \)**:
\[ x = 1 - \sqrt{2} \]
To find \( \frac{1}{x} \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1}{1 - \sqrt{2}} \cdot \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{1 + \sqrt{2}}{(1 - \sqrt{2})(1 + \sqrt{2})} \]
The denominator simplifies as follows:
\[ (1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1 \]
Thus,
\[ \frac{1}{x} = \frac{1 + \sqrt{2}}{-1} = -1 - \sqrt{2} \]

2. **Calculate \( x - \frac{1}{x} \)**:
Now we can calculate \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = (1 - \sqrt{2}) - (-1 - \sqrt{2}) = (1 - \sqrt{2}) + (1 + \sqrt{2}) = 1 - \sqrt{2} + 1 + \sqrt{2} = 2 \]

3. **Calculate \( (x - \frac{1}{x})^3 \)**:
Now we need to find \( (x - \frac{1}{x})^3 \):
\[ (x - \frac{1}{x})^3 = 2^3 = 8 \]

### Final Answer:
The value of \( (x - \frac{1}{x})^3 \) is \( \boxed{8} \).

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