Algebra and Functions

This lesson covers Algebra and Functions as required by the A-Level Mathematics Pure 1 specification. This topic forms the foundation for almost every other area of pure mathematics. You must be confident with algebraic manipulation, the laws of indices, surds, quadratic functions, simultaneous equations, inequalities, polynomial division, the factor theorem, and graph transformations.

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Laws of Indices

The laws of indices allow you to simplify expressions involving powers. Let a > 0 and m, n be rational numbers:

Rule	Law
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power	(aᵐ)ⁿ = aᵐⁿ
Zero index	a⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ
Fractional index	a^(1/n) = ⁿ√a
Combined fractional	a^(m/n) = (ⁿ√a)ᵐ

Example: Simplify 8^(2/3) × 2⁻⁴.

8^(2/3) = (∛8)² = 2² = 4
So 8^(2/3) × 2⁻⁴ = 4 × 1/16 = 1/4

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Surds