Find the Value Find the value \[ x = 2 + \sqrt{3} \] \[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} = 2 – \sqrt{3} \] \[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 – \sqrt{3}) = 4 \] \[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 – 3(x […]

Simplify Expression Simplify \[ \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} – \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} – \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} \] Solution: \[ \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} \times \frac{\sqrt{10} – \sqrt{3}}{\sqrt{10} – \sqrt{3}} = \frac{7\sqrt{3}(\sqrt{10} – \sqrt{3})}{10 – 3} = \sqrt{3}(\sqrt{10} – \sqrt{3}) \] \[ = \sqrt{30} – 3 \] \[ \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} – \sqrt{5}}{\sqrt{6} – \sqrt{5}} =

Simplify Expression Simplify \[ \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} – \frac{7 – 3\sqrt{5}}{3 – \sqrt{5}} \] Solution: \[ \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} \times \frac{3 – \sqrt{5}}{3 – \sqrt{5}} \] \[ = \frac{(7 + 3\sqrt{5})(3 – \sqrt{5})}{9 – 5} \] \[ = \frac{21 – 7\sqrt{5} + 9\sqrt{5} – 15}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3

Simplify Expression Simplify \[ \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} – \sqrt{2}} \] Solution: \[ \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} \times \frac{3\sqrt{2} – 2\sqrt{3}}{3\sqrt{2} – 2\sqrt{3}} \] \[ = \frac{(3\sqrt{2} – 2\sqrt{3})^2}{(3\sqrt{2})^2 – (2\sqrt{3})^2} \] \[ = \frac{18 – 12\sqrt{6} + 12}{18 – 12} \] \[ = \frac{30 – 12\sqrt{6}}{6} = 5 – 2\sqrt{6}

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: \[ \frac{4 + 3\sqrt{5}}{4 – 3\sqrt{5}} \times \frac{4 + 3\sqrt{5}}{4 + 3\sqrt{5}} \] \[ = \frac{(4 + 3\sqrt{5})^2}{4^2 – (3\sqrt{5})^2} \] \[ = \frac{16 + 24\sqrt{5} + 45}{16 – 45} \] \[ =