Surd - SS3 Mathematics Past Questions and Answers - page 1

$$8.\ddot{37}=8.373737\dots$$

Let this equal $x$

$x=8.\ddot{37}=8.373737\dots$ [1]

Multiply both sides by 100

$100x=837.3737\dots$ [2]

Subtract [1] from [2]

$$99x=829$$

$x=\frac{829}{99}$

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

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

$$3\sqrt{2}\times \sqrt{18}\times 5\sqrt{2}=3\sqrt{2}\times \sqrt{9\times 2}\times 5\sqrt{2}=3\sqrt{2}\times 3\sqrt{2}\times 5\sqrt{2}$$

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

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

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

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

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