Multiple Choice

1. Which of the following numbers is a surd?

   $\sqrt{169}$

   $\sqrt{0.25}$

   $\sqrt{18}$

   $\frac{7}{3}$

2. Which of the following statements about surds is INCORRECT?

   The product of two surds can be a rational number.

   Every surd is an irrational number.

   The square root of any prime number is a surd.

   For any positive integers $a$ and $b$, $\sqrt{a} + \sqrt{b} = \sqrt{a+b}$.

3. Simplify the surd expression $\sqrt{72x^5y^8}$, where $x > 0$ and $y > 0$.

   $6x^3y^4\sqrt{2}$

   $36x^2y^4\sqrt{2x}$

   $6x^2y^8\sqrt{2x}$

   $6x^2y^4\sqrt{2x}$

4. Simplify the expression $\sqrt{128a^3b^7c^2}$, where $a, b, c$ are positive variables.

   $8ab^3c\sqrt{2ab}$

   $8ab^3c\sqrt{ab}$

   $64ab^3c\sqrt{2ab}$

   $8a^2b^3c\sqrt{2ab}$

5. Simplify the expression $3\sqrt{45x^7}$, where $x > 0$.

   $9x^7\sqrt{5}$

   $9x^3\sqrt{5x}$

   $3x^3\sqrt{45x}$

   $9x^3\sqrt{5x^2}$

6. Simplify the expression $2\sqrt{12} + \sqrt{75} - 3\sqrt{27} + \sqrt{48} - \sqrt{3}$.

   $4\sqrt{3}$

   $5\sqrt{3}$

   $2\sqrt{3}$

   $3\sqrt{3}$

7. Simplify the expression $\sqrt{18} + \sqrt{50} - \sqrt{8} + \sqrt{98} - \sqrt{32}$.

   $11\sqrt{2}$

   $10\sqrt{2}$

   $8\sqrt{2}$

   $9\sqrt{2}$

8. Simplify the expression $3\sqrt{20} - 2\sqrt{45} + \sqrt{80} - 4\sqrt{5} + \sqrt{125}$.

   $4\sqrt{5}$

   $3\sqrt{5}$

   $6\sqrt{5}$

   $5\sqrt{5}$

9. Simplify the expression $5\sqrt{24} - 2\sqrt{54} + 3\sqrt{150} - \sqrt{6} + 2\sqrt{96}$.

   $26\sqrt{6}$

   $24\sqrt{6}$

   $25\sqrt{6}$

   $27\sqrt{6}$

10. Simplify the expression $\sqrt{108} + 2\sqrt{75} - \sqrt{243} + 3\sqrt{12} - \sqrt{300}$.

    $3\sqrt{3}$

    $6\sqrt{3}$

    $4\sqrt{3}$

    $5\sqrt{3}$

11. Simplify the product $(2\sqrt{3})(5\sqrt{6})$.

    $20\sqrt{3}$

    $60\sqrt{2}$

    $10\sqrt{18}$

    $30\sqrt{2}$

12. Expand and simplify $(3 + \sqrt{2})(4 - \sqrt{2})$.

    $10 - \sqrt{2}$

    $10 + \sqrt{2}$

    $10 + 2\sqrt{2}$

    $12 - 2\sqrt{2}$

13. Expand and simplify $(5\sqrt{3} - 2)(2\sqrt{3} + 1)$.

    $28 + \sqrt{3}$

    $28 - 2\sqrt{3}$

    $26 + \sqrt{3}$

    $28 - \sqrt{3}$

14. Simplify $(7 - 2\sqrt{5})(7 + 2\sqrt{5})$.

    $29$

    $49 - 2\sqrt{5}$

    $39$

    $49 - 20\sqrt{5}$

15. Simplify $(3\sqrt{2} + \sqrt{5})^2$.

    $18 + 5\sqrt{10}$

    $19 + 6\sqrt{10}$

    $23 + 6\sqrt{10}$

    $23 + \sqrt{10}$

16. Simplify the product $(\sqrt{8x})(\sqrt{2x^3})$, where $x > 0$.

    $4x^2$

    $16x^2$

    $4x^3$

    $4x^2\sqrt{x}$

17. Expand and simplify $(3\sqrt{5} - \sqrt{2})(2\sqrt{5} + 3\sqrt{2})$.

    $24 + 5\sqrt{10}$

    $34 + 7\sqrt{10}$

    $24 - \sqrt{10}$

    $24 + 7\sqrt{10}$

18. Simplify the expression $\frac{\sqrt{72}}{\sqrt{8}}$.

    $2$

    $3$

    $3\sqrt{2}$

    $2\sqrt{3}$

19. Rationalise the denominator of $\frac{6}{\sqrt{18}}$.

    $\frac{\sqrt{2}}{3}$

    $2$

    $\sqrt{2}$

    $\frac{6\sqrt{18}}{18}$

20. Rationalise the denominator of $\frac{3x}{\sqrt{12x^3}}$, where $x > 0$.

    $\frac{\sqrt{3}}{2x}$

    $\frac{1}{2x\sqrt{3x}}$

    $\frac{\sqrt{3x}}{2x}$

    $\frac{\sqrt{3x}}{2}$

21. Rationalise the denominator of $\frac{1}{3 + \sqrt{2}}$.

    $\frac{3 + \sqrt{2}}{11}$

    $\frac{3 - \sqrt{2}}{7}$

    $\frac{3 - \sqrt{2}}{11}$

    $\frac{3 + \sqrt{2}}{7}$

22. Rationalise the denominator of $\frac{\sqrt{3}}{2\sqrt{3} - \sqrt{6}}$.

    $\frac{6 + \sqrt{18}}{6}$

    $\frac{1 + \sqrt{2}}{2}$

    $1 - \frac{\sqrt{2}}{2}$

    $\frac{2 + \sqrt{2}}{2}$

23. Rationalise the denominator and simplify $\frac{5 + \sqrt{5}}{5 - \sqrt{5}}$.

    $2 + \sqrt{5}$

    $\frac{3 - \sqrt{5}}{2}$

    $\frac{15 + 10\sqrt{5}}{20}$

    $\frac{3 + \sqrt{5}}{2}$

24. Simplify the expression $\frac{10}{\sqrt{5}} + \sqrt{20} - \frac{1}{\sqrt{5} - \sqrt{3}}$.

    $\frac{6\sqrt{5} - \sqrt{3}}{2}$

    $4\sqrt{5} - \frac{\sqrt{5} + \sqrt{3}}{2}$

    $\frac{7\sqrt{5} - \sqrt{3}}{2}$

    $\frac{7\sqrt{5} + \sqrt{3}}{2}$

25. Simplify the expression $\frac{\sqrt{48} - \sqrt{12}}{\sqrt{3}} + \frac{1}{\sqrt{3} - 1}$.

    $1 + \sqrt{3}$

    $3 + \frac{\sqrt{3} + 1}{2}$

    $\frac{5 + \sqrt{3}}{2}$

    $\frac{4 + \sqrt{3}}{2}$

26. Given $x = \frac{1}{2-\sqrt{3}}$ and $y = \frac{1}{2+\sqrt{3}}$, determine the value of $x^2 + y^2$.

    $10$

    $16$

    $12$

    $14$

27. If $P = (\sqrt{5} + \sqrt{3})^2$ and $Q = (\sqrt{5} - \sqrt{3})^2$, find the value of $\frac{P+Q}{P-Q}$.

    $4\sqrt{15}$

    $\frac{\sqrt{15}}{4}$

    $\frac{4\sqrt{15}}{15}$

    $\frac{8\sqrt{15}}{15}$