If `a=1-sqrt(2),"then find the value of "(a-(1)/(a))^(3)`

Explanation

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

### Step 1: Find \( \frac{1}{a} \)
Given \( a = 1 - \sqrt{2} \), we first need to calculate \( \frac{1}{a} \).

\[ \frac{1}{a} = \frac{1}{1 - \sqrt{2}} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 1 + \sqrt{2} \):

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

### Step 2: Calculate \( a - \frac{1}{a} \)
Now we can find \( a - \frac{1}{a} \):

\[ a - \frac{1}{a} = (1 - \sqrt{2}) - (-(1 + \sqrt{2})) \]

This simplifies to:

\[ = (1 - \sqrt{2}) + (1 + \sqrt{2}) = 1 - \sqrt{2} + 1 + \sqrt{2} = 2 \]

### Step 3: Calculate \( (a - \frac{1}{a})^3 \)
Now we need to compute \( (a - \frac{1}{a})^3 \):

\[ (a - \frac{1}{a})^3 = 2^3 = 8 \]

### Final Answer
Thus, the value of \( (a - \frac{1}{a})^3 \) is \( 8 \).

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