Indices, roots, standard form and exact calculation Study Guide

A surd is the root of a number that cannot be simplified to a whole number. Examples include \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{7}\). Numbers like \(\sqrt{4}\) and \(\sqrt{9}\) are not surds because they simplify to integers (2 and 3, respectively).

Manipulating surds involves applying rules similar to those for algebraic expressions. The fundamental properties are:

1. Multiplication: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\). For example, \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\).
2. Division: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\, (b \neq 0)\). For example, \(\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{5}\).

Simplifying surds involves finding the largest square factor of the number under the root sign. For example, to simplify \(\sqrt{12}\), we find that 4 is the largest square factor of 12. So, \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).

Adding and subtracting surds is only possible when the surds are 'like surds', meaning they have the same number under the root sign. For example, \(3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}\). If surds are not alike, they must first be simplified to see if they can become like surds. For example, \(\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\).

Rationalising the denominator means removing any surds from the denominator of a fraction. If the denominator is a single surd, multiply the numerator and denominator by that surd. For example, \(\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). If the denominator is of the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), multiply the numerator and denominator by its conjugate. The conjugate of \(a + \sqrt{b}\) is \(a - \sqrt{b}\), and vice versa. This uses the difference of two squares identity: \((x+y)(x-y) = x^2 - y^2\). For example, to rationalise \(\frac{1}{2 + \sqrt{3}}\), we multiply by \(\frac{2 - \sqrt{3}}{2 - \sqrt{3}}\): \(\frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2)^2 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3}\).

These higher methods allow for the simplification of complex expressions involving surds and are essential for solving related problems in algebra and geometry.