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

Explanation

To solve the problem step by step, we will find the values for each part as follows:

Given:
\[ x = 2 + \sqrt{3} \]

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

1. Start with the expression:
\[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \]
2. To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \]
3. Calculate the denominator using the difference of squares:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
4. Therefore, we have:
\[ \frac{1}{x} = 2 - \sqrt{3} \]

### (ii) Find \( x + \frac{1}{x} \)

1. Substitute the values of \( x \) and \( \frac{1}{x} \):
\[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 - \sqrt{3}) \]
2. Combine like terms:
\[ = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]

### (iii) Find \( x - \frac{1}{x} \)

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

### (iv) Find \( x^2 + \frac{1}{x^2} \)

1. Use the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
2. Substitute the value of \( x + \frac{1}{x} \):
\[ x^2 + \frac{1}{x^2} = 4^2 - 2 = 16 - 2 = 14 \]

### Summary of Results:
1. \( \frac{1}{x} = 2 - \sqrt{3} \)
2. \( x + \frac{1}{x} = 4 \)
3. \( x - \frac{1}{x} = 2\sqrt{3} \)
4. \( x^2 + \frac{1}{x^2} = 14 \)