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

Options

A. 0
B. `- sqrt(2)`
C. `-2`
D. `3 sqrt(2)`

Answer

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

### Step-by-Step Solution:

1. **Express \( x^3 \)**:
\[ x = (\sqrt{2} + 1)^{-\frac{1}{3}} \]
To find \( x^3 \), we cube both sides:
\[ x^3 = \left((\sqrt{2} + 1)^{-\frac{1}{3}}\right)^3 = (\sqrt{2} + 1)^{-1} \]

2. **Rewrite \( x^3 \)**:
\[ x^3 = \frac{1}{\sqrt{2} + 1} \]

3. **Rationalize \( x^3 \)**:
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} - 1 \):
\[ x^3 = \frac{1}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1 \]

4. **Find \( \frac{1}{x^3} \)**:
Since \( x^3 = \sqrt{2} - 1 \), we can find \( \frac{1}{x^3} \):
\[ \frac{1}{x^3} = \frac{1}{\sqrt{2} - 1} \]
Again, we rationalize the denominator:
\[ \frac{1}{x^3} = \frac{1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{2} + 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} + 1}{2 - 1} = \sqrt{2} + 1 \]

5. **Calculate \( x^3 - \frac{1}{x^3} \)**:
Now we can substitute \( x^3 \) and \( \frac{1}{x^3} \) into the expression:
\[ x^3 - \frac{1}{x^3} = (\sqrt{2} - 1) - (\sqrt{2} + 1) \]
Simplifying this gives:
\[ x^3 - \frac{1}{x^3} = \sqrt{2} - 1 - \sqrt{2} - 1 = -2 \]

### Final Answer:
The value of \( x^3 - \frac{1}{x^3} \) is \( -2 \).

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