IP EMaths Notes (Upper Sec, Year 3-4): 03) Algebraic Fractions and Surds

Exact manipulation avoids rounding drift. Reduce algebraic fractions fully before substituting values, and rationalise surd denominators when presenting final answers.

Key techniques

• Factor numerator and denominator before cancelling: \(\frac{x^{2} - 9}{x^{2} - x - 6} = \frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}\).
• State excluded values after cancellation (here, \(x \neq 3\) and \(x \neq -2\)).
• Rationalise denominators with conjugates: multiply \(\dfrac{1}{a + \sqrt{b}}\) by \(\dfrac{a - \sqrt{b}}{a - \sqrt{b}}\).

Worked example - Simplify and rationalise

Simplify \(\dfrac{2x^{2} - 8}{x^{2} - 4x + 4}\) and state any restrictions, then rationalise \(\dfrac{3}{\sqrt{5} - 1}\).

1. Factor numerator: \(2x^{2} - 8 = 2(x^{2} - 4) = 2(x - 2)(x + 2)\).
2. Factor denominator: \(x^{2} - 4x + 4 = (x - 2)^{2}\).
3. Cancel a common \((x - 2)\): \(\dfrac{2(x - 2)(x + 2)}{(x - 2)^{2}} = \dfrac{2(x + 2)}{x - 2}\).
4. Restrictions: \((x - 2)^{2} \neq 0 \Rightarrow x \neq 2\).
5. Rationalise \(\dfrac{3}{\sqrt{5} - 1}\): multiply by \(\dfrac{\sqrt{5} + 1}{\sqrt{5} + 1}\).
6. Numerator: \(3(\sqrt{5} + 1)\). Denominator: \((\sqrt{5})^{2} - 1^{2} = 5 - 1 = 4\).
7. Simplified: \(\dfrac{3\sqrt{5} + 3}{4}\).

Try this

Given \(x > 0\), simplify \(\dfrac{\sqrt{18x}}{\sqrt{2x} + \sqrt{8x}}\) and present the answer with a rational denominator.