Use surds and the laws of indices: simplify surds, rationalise denominators, and apply the index laws including zero, negative and fractional indices.

What this dot point is asking

Surds and indices give you exact arithmetic, and CCEA GCSE Further Mathematics expects total fluency with both. You must simplify surds, rationalise denominators (including the harder case with a two-term denominator), and use every index law, including zero, negative and fractional powers. These are non-calculator staples that feed straight into quadratics, the equation of a circle, logarithms and calculus, so accuracy with exact forms is rewarded throughout the Pure unit.

Simplifying and combining surds

To simplify a surd, find the largest perfect-square factor and take its root outside. For example $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$. You can only add or subtract surds when the part under the root matches, just like collecting like terms: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $\sqrt{2} + \sqrt{3}$ cannot be combined. To multiply, use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, then simplify; remember $\sqrt{a} \times \sqrt{a} = a$.

Rationalising denominators

A fraction is not in its simplest exact form while a surd sits in the denominator, so we rationalise it. For a single surd, multiply numerator and denominator by that surd: $\dfrac{5}{\sqrt{2}} = \dfrac{5\sqrt{2}}{2}$. For a two-term denominator, multiply by the conjugate, which uses the difference of two squares to remove the root entirely.

The laws of indices

The index laws apply to any base, with negative and fractional powers extending them to the whole number line. Combine the laws in sequence rather than all at once.

When evaluating something like $27^{\frac{2}{3}}$, take the cube root first ($27^{\frac{1}{3}} = 3$), then square it ($3^2 = 9$). Doing the root before the power keeps the numbers small. For a negative fractional index, deal with the reciprocal first, then the root and power.

Indices in algebra

The same laws apply to algebraic terms. For instance $\dfrac{6x^5}{2x^2} = 3x^3$, and $(2x^3)^4 = 2^4 x^{12} = 16x^{12}$. Writing roots and reciprocals as powers, such as $\sqrt{x} = x^{\frac{1}{2}}$ and $\dfrac{1}{x^3} = x^{-3}$, is essential before you differentiate or integrate later in the unit, because the calculus rules need every term in index form first.

Why this matters

Exact arithmetic with surds and full command of indices run right through Pure. Surds appear in the quadratic formula, in the equation of a circle and in trigonometric exact values; index laws are the gateway to logarithms and to rewriting terms so they can be differentiated or integrated. Because so many of these questions are non-calculator, fluency here protects marks across the unit.