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

Answer

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

### Step-by-Step Solution:

1. **Calculate \(x - \frac{1}{x}\):**
\[ x - \frac{1}{x} = 1 - \sqrt{2} - \frac{1}{1 - \sqrt{2}} \]

2. **Rationalize the denominator of \(\frac{1}{1 - \sqrt{2}}\):**
\[ \frac{1}{1 - \sqrt{2}} \times \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{1 + \sqrt{2}}{(1 - \sqrt{2})(1 + \sqrt{2})} \]

3. **Simplify the denominator using the difference of squares formula:**
\[ (1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1 \]
So,
\[ \frac{1 + \sqrt{2}}{-1} = -(1 + \sqrt{2}) = -1 - \sqrt{2} \]

4. **Substitute back into the expression:**
\[ x - \frac{1}{x} = 1 - \sqrt{2} - (-1 - \sqrt{2}) = 1 - \sqrt{2} + 1 + \sqrt{2} \]
Simplify:
\[ x - \frac{1}{x} = (1 + 1) + (-\sqrt{2} + \sqrt{2}) = 2 \]

5. **Calculate \((x - \frac{1}{x})^3\):**
\[ (x - \frac{1}{x})^3 = 2^3 = 8 \]

### Final Answer:
\[ (x - \frac{1}{x})^3 = 8 \]