Laws of Indices & Surds (AQA A-Level Mathematics): Revision Notes

Example: Write $\frac{14}{\sqrt{7}}$ in simplified surd form

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

• Like writing $\frac{14\sqrt{7}}{7}$ which can be simplified.

Example: Write $\frac{7}{5\sqrt{6}}$ in simplified surd form

$\frac{7}{5\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{7\sqrt{6}}{5 \times 6} = \frac{7\sqrt{6}}{30}$

• This method ensures correct simplification.
• Incorrect Method (for learning purposes):

$\frac{7}{5\sqrt{6}} \times \frac{5\sqrt{6}}{5\sqrt{6}} = \frac{35\sqrt{6}}{150} = \frac{7\sqrt{6}}{30}$

Example: Simplify $\frac{\sqrt{45} - 5}{\sqrt{20}}$:

• Hint: Any surds that can be simplified should be first to make the solution as easy as possible.

$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

$\frac{3\sqrt{5} - 5}{2\sqrt{5}} = \frac{\sqrt{5}(3 - \sqrt{5})}{2\sqrt{5}} = \frac{15 - 5\sqrt{5}}{10} = \frac{3 - \sqrt{5}}{2}$

Example: Express $\frac{3 + \sqrt{20}}{3 + \sqrt{5}}$ in the form $a + b\sqrt{5}$ :

$\frac{3 + \sqrt{20}}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{(3 + 2\sqrt{5})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}$

$= \frac{3(3) + 3(-\sqrt{5}) + 2\sqrt{5}(3) - 2\sqrt{5}(\sqrt{5})}{9 - 5}$

$= \frac{9 - 3\sqrt{5} + 6\sqrt{5} - 10}{4}$

$= \frac{-1 + 3\sqrt{5}}{4}$

$= -\frac{1}{4} + \frac{3\sqrt{5}}{4}$

Get into the habit of rationalising the denominator! It helps you build up that skill by the time you sit your exams.