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

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

3. If \(x=\sqrt3+\sqrt2\), then find the simplest value of \(x-\cfrac{1}{x}\).

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

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

6. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x - \frac{1}{x} \]

7. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x + \frac{1}{x} \]

8. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x^2 + \frac{1}{x^2} \]

9. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x^3 + \frac{1}{x^3} \]

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

11. If \((5 + √3)(5 - √3) = 25 - x^2\), find the value of \(x\).

(a) 3 (b) √3 (c) -√3 (d) \(\pm\)√3

12. 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) \(\cfrac{1}{2709}\)

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

(a) 199 (b) 195 (c) 198 (d) 201

14. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

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

16. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

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

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

19. If \(x + \frac{9}{x} = 6\), find the numerical value of \(x^2\).

20. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

21. \((5 + \sqrt{3})(5 - \sqrt{3}) = 25 - x^2\); find the value of \(x\).

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

23. Find the value of \(x\) from the equation: \(x \sec^2 45^\circ \cdot \csc^2 45^\circ + 2(\sin 60^\circ + \sin 30^\circ) = \tan 60^\circ\)

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

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

26. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

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

28. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

29. If \(x = \sqrt{2} + 1\), find the values of \(x^4 + \frac{1}{x^4}\) and \(x^4 - \frac{1}{x^4}\).

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