If `x=3+2sqrt(2)`, find :
`(i) (1)/(x)` `(ii) x-(1)/(x)`
`(iii) (x-(1)/(x))^(3)` `(iv)x^(3)-(1)/(x^(3))`

Answer

Let's solve the problem step by step.

Given:
\( x = 3 + 2\sqrt{2} \)

### Part (i): Find \( \frac{1}{x} \)

1. Start with the expression for \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \]

2. To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1 \cdot (3 - 2\sqrt{2})}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]

3. Calculate the denominator using the difference of squares:
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]

4. Thus,
\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

### Part (ii): Find \( x - \frac{1}{x} \)

1. Substitute the values of \( x \) and \( \frac{1}{x} \):
\[ x - \frac{1}{x} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) \]

2. Simplify:
\[ x - \frac{1}{x} = 3 + 2\sqrt{2} - 3 + 2\sqrt{2} = 4\sqrt{2} \]

### Part (iii): Find \( (x - \frac{1}{x})^3 \)

1. Use the result from Part (ii):
\[ (x - \frac{1}{x})^3 = (4\sqrt{2})^3 \]

2. Calculate:
\[ (4\sqrt{2})^3 = 4^3 \cdot (\sqrt{2})^3 = 64 \cdot 2\sqrt{2} = 128\sqrt{2} \]

### Part (iv): Find \( x^3 - \frac{1}{x^3} \)

1. Use the identity:
\[ x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x - \frac{1}{x}) \]

2. Substitute the values from Part (ii) and Part (iii):
\[ x^3 - \frac{1}{x^3} = 128\sqrt{2} + 3(4\sqrt{2}) \]

3. Simplify:
\[ x^3 - \frac{1}{x^3} = 128\sqrt{2} + 12\sqrt{2} = 140\sqrt{2} \]

### Summary of Answers:
- (i) \( \frac{1}{x} = 3 - 2\sqrt{2} \)
- (ii) \( x - \frac{1}{x} = 4\sqrt{2} \)
- (iii) \( (x - \frac{1}{x})^3 = 128\sqrt{2} \)
- (iv) \( x^3 - \frac{1}{x^3} = 140\sqrt{2} \)