Surds and Indices in 0606: The Assumed Knowledge You Still Need

In the 2025-2027 Cambridge IGCSE Additional Mathematics (0606) syllabus, indices and surds are assumed knowledge, not taught content. That is easy to misread as “not examined”. It is the opposite: they are assumed because they turn up everywhere, especially on the non-calculator Paper 1, where exact answers in surd form are the norm. This guide is the foundation to have automatic before the rest of the course leans on it.

The index laws to have fluent

These should be instant, in both directions:

• $a^m \times a^n = a^{m+n}$, and $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$, and $a^{-n} = \dfrac{1}{a^n}$
• $a^{1/n} = \sqrt[n]{a}$, and $a^{m/n} = \sqrt[n]{a^m}$

The most common 0606 slip is fumbling a negative or fractional index under time pressure, for example in a differentiation question where you first rewrite $\dfrac{1}{\sqrt{x}}$ as $x^{-1/2}$ before applying the rule. If that rewrite is not automatic, the calculus that follows inherits the delay.

The surd rules to have fluent

• $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$, so $\sqrt{50} = \sqrt{25}\,\sqrt{2} = 5\sqrt{2}$
• $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$
• Rationalising the denominator: multiply by the conjugate. $\dfrac{1}{2+\sqrt{3}} = \dfrac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})} = \dfrac{2-\sqrt{3}}{1} = 2-\sqrt{3}$

On Paper 1 you cannot reach for a decimal, so “leave your answer in exact form” means a simplified surd. A quadratic solved by formula with a non-square discriminant ends in surd form, and the marks are for the simplified version.

Where this shows up in 0606

• Non-calculator Paper 1: exact answers throughout, from quadratics to coordinate geometry distances.
• Exact trigonometric values: $\sin 60° = \dfrac{\sqrt{3}}{2}$ and its family, used in trigonometry and circular measure.
• Calculus: results and stationary-point coordinates often left in exact, surd or index form.

If this is shaky

A gap here is quiet, it does not announce itself as a “surds question”, it just slows everything down and invites careless errors. That makes it a high-return thing to fix early. The readiness check will tell you fast whether it needs work, and a 1-hour trial class can drill it against real exam questions so it stops costing marks.