Surd - SS3 Mathematics Past Questions and Answers - page 1

\[8.\ddot{37} = 8.373737\ldots\]

Let this equal \(x\)

\(x = 8.\ddot{37} = 8.373737\ldots\) [1]

Multiply both sides by 100

\(100x = 837.3737\ldots\) [2]

Subtract [1] from [2]

\[99x = 829\]

\(x = \frac{829}{99}\)

\[\frac{\sqrt{28}}{2\sqrt{35}} = \ \frac{\sqrt{4 \times 7}}{2\sqrt{35}} = \frac{2\sqrt{7}}{2\sqrt{35}} = \frac{2\sqrt{7}}{2\sqrt{35}}.\frac{\sqrt{35}}{\sqrt{35}} = \ \frac{2\sqrt{7 \times 35}}{2 \times 35} = \frac{2\sqrt{7 \times 7 \times 5}}{70} = \frac{2\sqrt{49 \times 5}}{70}\]

\(= \frac{2 \times 7\sqrt{5}}{70} = \frac{14\sqrt{5}}{70} = \frac{\sqrt{5}}{5}\)

\[3\sqrt{2} \times \sqrt{18} \times 5\sqrt{2} = \ 3\sqrt{2} \times \sqrt{9 \times 2} \times 5\sqrt{2} = \ 3\sqrt{2} \times 3\sqrt{2} \times 5\sqrt{2}\]

\(9 \times \left( \sqrt{2} \right)^{2} \times 5\sqrt{2} = 9 \times 2 \times 5\sqrt{2} = 18 \times 5\sqrt{2} = 90\sqrt{2}\)

\[\sqrt{108} - \sqrt{75} = \ \sqrt{36 \times 3} - \sqrt{25 \times 3} = \ 6\sqrt{3} - 5\sqrt{3} = \sqrt{3}\]

\[\frac{2}{\sqrt{2} + \sqrt{5}} = \frac{2}{\sqrt{2} + \sqrt{5}}.\frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} - \sqrt{5}} = \frac{2(\sqrt{2} - \sqrt{5})}{2 - 5} = \frac{2(\sqrt{2} - \sqrt{5})}{- 3} = \frac{- 2(\sqrt{2} - \sqrt{5})}{3}\]

\(\frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} + \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} = \ \frac{\left( \sqrt{5} - \sqrt{3} \right)\left( 1 + \sqrt{2} \right) + (\sqrt{5} + \sqrt{3})(1 - \sqrt{2})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} = \frac{\left( \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} \right) + (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6})}{5 - 3}\)

\(\frac{\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} + \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}}{2} = \ \frac{2\sqrt{5} - 2\sqrt{6}}{2} = \frac{2(\sqrt{5} - \sqrt{6})}{2} = \sqrt{5} - \sqrt{6}\)