Algebra and Functions (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
Test yourself on Algebra and Functions (AS Unit 1: Pure Mathematics A) with WJEC A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
Algebra and Functions (AS Unit 1: Pure Mathematics A) explained
Indices govern powers and roots through three core laws for a common base a > 0: multiplication adds indices, aᵖ × aᵠ = a^(p+q); division subtracts them, aᵖ ÷ aᵠ = a^(p−q); and a power of a power multiplies them, (aᵖ)ᵠ = a^(pq).
Read the full explanation
Negative indices give reciprocals, a⁻ᵖ = 1/aᵖ, and a⁰ = 1 for a ≠ 0. Fractional indices link to radicals: a^(1/q) = ᑫ√a, so a^(p/q) = (ᑫ√a)ᵖ = ᑫ√(aᵖ); for example 8^(2/3) = (³√8)² = 4. Simplifying algebraic expressions such as (2x^(1/2) + 3x^(−1/2))² requires systematic expansion, adding powers of x term by term. Surds may be converted to fractional exponents to simplify manipulation, but this is a convenience, not a prerequisite: surds can also be differentiated or integrated directly.
Your focus
- Apply the laws of indices to simplify complex algebraic and numerical expressions.
- Convert between radical notation, reciprocal expressions, and rational exponents.
- Manipulate expressions with fractional and negative powers to solve equations.
Algebra and Functions (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- Converting surds and reciprocals into index form with fractional or negative exponents, e.g. √x = x^(1/2) and 1/x³ = x⁻³.
- Applying index laws correctly when multiplying or dividing powers of the same base, adding or subtracting indices rather than multiplying them.
- Evaluating numerical rational powers by taking the root first, e.g. 27^(2/3) = (³√27)² = 9, and checking the sign where the base is negative.
- Expanding brackets involving fractional powers, such as (2x^(1/2) + 3x^(−1/2))², and collecting like terms by adding indices accurately.
- Simplifying compound expressions by combining several laws in sequence, e.g. (4x⁶)^(1/2) ÷ x² = 2x.
Examiner Tips
- 💡Rewrite all roots and reciprocals in index notation before simplifying or differentiating; this makes the laws of indices directly applicable and reduces sign errors.
- 💡Check each step against the three core laws, and remember a⁰ = 1 (not 0) for any a ≠ 0.
Common Mistakes
- Multiplying powers instead of adding them when multiplying terms with the same base, e.g. writing x² × x³ = x⁶; the correct result is x⁵.
- Confusing a negative power with a negative coefficient, e.g. writing x⁻² = −x²; the correct meaning is x⁻² = 1/x².
- Misinterpreting fractional powers, e.g. treating 8^(1/3) as 8/3; the correct value is ³√8 = 2.
- Applying a power to only part of a product, e.g. writing (2x)³ = 2x³; the correct expansion is 8x³.