GCSE Maths: Surds and Indices (Higher)
Surds and indices are Higher tier territory in GCSE Mathematics, and they reward precise rule-following more than almost any other topic: one sign slip or misapplied law and the whole answer collapses. This quiz drills the two topics together, exactly as the specification groups them, because fractional indices are what connect powers to roots.
On the indices side, you will apply the laws for multiplying and dividing powers, raise powers to powers, and evaluate expressions with negative and fractional exponents — including classics like 8^(2/3) and 16^(3/4), which test whether you take the root first and then the power. One question deliberately chains three laws in a single expression, finishing at x⁰ = 1, the case students most often second-guess.
On the surds side, the quiz covers simplifying products of roots, expanding brackets containing surds — both the difference-of-two-squares pattern (3 + √5)(3 − √5) that collapses to a rational number, and the full binomial square (2 + √3)² with its middle term — and rationalising denominators, first with a single surd and then with a binomial denominator requiring the conjugate.
Every question is written to be answerable without a calculator, matching the non-calculator paper where these skills are usually assessed, and each explanation shows the complete manipulation step by step. The distractors are built from the genuine misconceptions: forgetting the middle term when squaring a binomial, adding instead of multiplying exponents, and rationalising with the wrong conjugate. If you are targeting grades 7–9, fluency here feeds directly into A-Level algebra, where these manipulations are assumed knowledge from the first lesson.
- Apply the laws of indices, including negative and fractional exponents
- Evaluate expressions such as 8^(2/3) without a calculator by taking the root before the power
- Simplify products of surds and recognise square factors
- Expand brackets containing surds, including conjugate pairs and binomial squares
- Rationalise denominators with single-surd and binomial denominators using the conjugate
Topic scope follows the Higher tier content of the Department for Education's GCSE mathematics subject content: calculating with fractional indices, exact calculation with surds, simplifying surd expressions and rationalising denominators.