1. If \(x = \sqrt{3} + \sqrt{2}\), then the value of \(x^3 + \frac{1}{x^3}\) is —

(a) \(18\sqrt2\) (b) \(18\sqrt3\) (c) \(18\sqrt5\) (d) \(18\sqrt6\)

2. If \(x = \sqrt{3} + \sqrt{2}\), then find the simplest value of \[ x^3 + \frac{1}{x^3} \]

3. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{x^3}{x^6 + 7x^3 + 1}\).

(a) \(\cfrac{1}{2737}\) (b) \(\cfrac{1}{2730}\) (c) \(\cfrac{1}{2710}\) (d) \(\cfrac{1}{2709}\)

4. If \(x = 3 + 2\sqrt{2}\), then find the value of \(\left(\sqrt{x} + \cfrac{1}{\sqrt{x}}\right)\).

5. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

6. Given \(x = 3 + \sqrt{3}\) and \(y = 6\), find the value of \((x + y)^2\).

7. \((5 + \sqrt{3})(5 - \sqrt{3}) = 25 - x^2\); find the value of \(x\).

8. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \cfrac{1}{\sqrt{3} + \sqrt{2}}\), then find the value of \((x + y)^2 + (x - y)^2\).

9. If \(x = \sqrt{3} + \frac{1}{\sqrt{3}}\) and \(y = \sqrt{3} - \frac{1}{\sqrt{3}}\), find the value of \(\frac{x^2}{y} + \frac{y^2}{x}\).

10. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then find the simplest value of \(\frac{x^2 - xy + y^2}{x^2 + xy + y^2}\).

11. If \(x = \frac{\sqrt{3}}{2}\), then find the value of \[ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}} \]

12. If \(x = 3 + 2\sqrt{2}\), find the value of \(x + \cfrac{1}{x}\).

13. If \(\cfrac{(5 + \sqrt{3})}{(5 - \sqrt{3})} = x - \sqrt{15}y\), then find the value of \(x + y\).

14. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), find the value of \(3x^2 + 5xy + 3y^2\).

15. If \(x = \sqrt{5} + 2\), determine the value of \(x^3 - \cfrac{1}{x^3}\).

16. If \(x = 7 + 4\sqrt{3}\), then find the value of \(x + \cfrac{1}{x}\).

17. If \(x=\sqrt3+\sqrt2\), then determine the simplified value of \(x^3-\cfrac{1}{x^3}\).

18. If \(x = 3 + 2\sqrt{2}\), then find the value of \(x + \cfrac{1}{x}\).

19. If \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 25 - x^2\) is an equation, find the value of \(x\).

20. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x - \frac{1}{x} \]

21. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ y^2 + \frac{1}{y^2} \]

22. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x^3 - \frac{1}{x^3} \]

23. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ xy + \frac{1}{xy} \]

24. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ 3x^2 - 5xy + 3y^2 \]

25. If \(x = 2 + \sqrt{3}\), then find the value of \[ x + \frac{1}{x} \]

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

26. If \(x = 3 + 2\sqrt{2}\), find the value of \(x + \cfrac{1}{x}\).

27. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

28. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{1}{\sqrt{x}}\).

29. If \( (\sqrt{3} - \sqrt{2})^x = (\sqrt{3} + \sqrt{2})^2 \), then find the value of \(x\).

(a) -2 (b) 1 (c) 2 (d) 3

30. If \(x = \cfrac{4\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then find the value of \[ \cfrac{x + \sqrt{20}}{x - \sqrt{20}} + \cfrac{x + \sqrt{12}}{x - \sqrt{12}}. \]