Sheet № 08 · Foundation + Higher · AQA · Edexcel · OCR
Indices and Index Laws –
Indices (also called powers or exponents) are a shorthand for repeated multiplication, and the index laws are the rules that let you simplify expressions involving them. This topic spans Foundation and Higher tiers: Foundation students need the basic laws for positive integer powers, while Higher students must also handle zero, negative,
§Key definitions
Question:
Simplify 3x⁴y² × 5x³y.
Q1 (Foundation):
Simplify p⁵ × p³ ÷ p².
Q2 (Higher):
Evaluate 16^(3/4).
Q3 (Higher):
Simplify (2x³)⁴.
§Formulas to memorise
a^m × a^n = a^(m+n) — when multiplying, add the powers
a^m ÷ a^n = a^(m−n) — when dividing, subtract the powers
(a^m)^n = a^(mn) — when raising a power to another power, multiply the powers
a^0 = 1 — any non-zero number to the power of zero equals 1
a^(−n) = 1/a^n — a negative power means the reciprocal
a^(1/n) = ⁿ√a — the nth root of a
a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m) — root first, then power (or vice versa)
Worked example
Simplify 3x⁴y² × 5x³y.
Working:
⚠ Common mistakes
- ✗Adding indices when you should multiply (and vice versa). Remember: × means add the powers, (a^m)^n means multiply the powers. These are different operations.
- ✗Applying index laws to different bases. 2³ × 3² cannot be simplified using index laws — the bases must match.
- ✗Thinking a^0 = 0. Any non-zero number to the power of zero is 1, not 0. This catches out many students.
- ✗Mishandling negative indices. 2⁻³ = 1/8, not −8. A negative index means reciprocal, not a negative number.
- ✗Taking the power before the root with fractional indices. While mathematically you can do either order, taking the root first keeps the numbers small. 64^(2/3): cube root of 64 is 4, then 4² = 16. If you cube first: 64² = 4096, then ∛4096 — much harder without a calculator.
✦ Exam tips
- →Write out each law you are using. When simplifying expressions, annotate your working: "multiplication law: add powers". This helps you avoid errors and can earn method marks.
- →Fractional indices almost always appear on non-calculator Higher papers. Practise evaluating values like 16^(3/4), 125^(2/3), and 32^(−3/5) until they are automatic. See our formulas guide for the key index law rules.
- →Look out for hidden index law questions in algebra. Simplifying algebraic fractions, solving equations, and working with surds all draw on these laws.
- →If stuck, rewrite everything using the definition. For example, x⁻¹ = 1/x and x^(1/2) = √x. This can make unfamiliar problems feel manageable.
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