May 2026

Find the Value Find the value correct to three decimal places \[ \frac{4}{3\sqrt{3} – 2\sqrt{2}} + \frac{3}{3\sqrt{3} + 2\sqrt{2}}, \quad \text{where } \sqrt{2} = 1.4142,\ \sqrt{3} = 1.732 \] Solution: \[ \frac{4(3\sqrt{3} + 2\sqrt{2}) + 3(3\sqrt{3} – 2\sqrt{2})}{(3\sqrt{3})^2 – (2\sqrt{2})^2} \] \[ = \frac{12\sqrt{3} + 8\sqrt{2} + 9\sqrt{3} – 6\sqrt{2}}{27 – 8} \] \[ = […]

Find the Value Find the value correct to three decimal places \[ \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}}, \quad \text{where } \sqrt{2} = 1.4142 \] Solution: \[ \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \] \[ = \frac{(1 + \sqrt{2})(3 + 2\sqrt{2})}{9 – 8} \] \[ = (1 + \sqrt{2})(3 + 2\sqrt{2})

Find the Value Find the value correct to three decimal places \[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}}, \quad \text{where } \sqrt{5} = 2.2360 \] Solution: \[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} \times \frac{3 – 2\sqrt{5}}{3 – 2\sqrt{5}} \] \[ = \frac{(3 – \sqrt{5})(3 – 2\sqrt{5})}{9 – 20} \] \[ = \frac{9 – 6\sqrt{5} – 3\sqrt{5}

Find the Value Find the value \[ \frac{6}{\sqrt{5} – \sqrt{3}}, \quad \text{where } \sqrt{5} = 2.236,\ \sqrt{3} = 1.732 \] Solution: \[ \frac{6}{\sqrt{5} – \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} \] \[ = \frac{6(\sqrt{5} + \sqrt{3})}{5 – 3} \] \[ = \frac{6(\sqrt{5} + \sqrt{3})}{2} \] \[ = 3(\sqrt{5} + \sqrt{3}) \] \[ = 3(2.236

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} = a\sqrt{5} + b \] Solution: \[ \frac{3 – \sqrt{5}}{3 + 2\sqrt{5}} \times \frac{3 – 2\sqrt{5}}{3 – 2\sqrt{5}} \] \[ = \frac{(3 – \sqrt{5})(3 – 2\sqrt{5})}{3^2 – (2\sqrt{5})^2} \] \[ = \frac{9 – 6\sqrt{5} – 3\sqrt{5} + 2 \cdot

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{4 + 3\sqrt{5}}{4 – 3\sqrt{5}} = a + b\sqrt{5} \] Solution: \[ \frac{4 + 3\sqrt{5}}{4 – 3\sqrt{5}} \times \frac{4 + 3\sqrt{5}}{4 + 3\sqrt{5}} \] \[ = \frac{(4 + 3\sqrt{5})^2}{4^2 – (3\sqrt{5})^2} \] \[ = \frac{16 + 24\sqrt{5} + 45}{16 – 45} \] \[ =

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} + \sqrt{7}} = a – b\sqrt{77} \] Solution: \[ \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} + \sqrt{7}} \times \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} – \sqrt{7}} \] \[ = \frac{(\sqrt{11} – \sqrt{7})^2}{11 – 7} \] \[ = \frac{11 + 7 – 2\sqrt{77}}{4} \] \[ = \frac{18 –

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{3 + \sqrt{2}}{3 – \sqrt{2}} = a + b\sqrt{2} \] Solution: \[ \frac{3 + \sqrt{2}}{3 – \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} \] \[ = \frac{(3 + \sqrt{2})^2}{3^2 – (\sqrt{2})^2} \] \[ = \frac{9 + 6\sqrt{2} + 2}{9 – 2} \] \[ =

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} = a – \sqrt{b} \] Solution: \[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} \times \frac{2 – \sqrt{2}}{2 – \sqrt{2}} \] \[ = \frac{(4 + \sqrt{2})(2 – \sqrt{2})}{(2)^2 – (\sqrt{2})^2} \] \[ = \frac{8 – 4\sqrt{2} + 2\sqrt{2} – 2}{4 –

Determine a and b Determine rational numbers \(a\) and \(b\) \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} = a – b\sqrt{3} \] Solution: \[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} – 1}{\sqrt{3} – 1} \] \[ = \frac{(\sqrt{3} – 1)^2}{(\sqrt{3})^2 – (1)^2} \] \[ = \frac{3 – 2\sqrt{3} + 1}{3 – 1} \] \[ =