10. Surds and Indices — Basics

Laws of Indices: - aᵐ × aⁿ = a^(m+n) - aᵐ ÷ aⁿ = a^(m−n) - (aᵐ)ⁿ = a^(mn) - a⁰ = 1 (for a ≠ 0) - a^(−n) = 1/aⁿ - a^(1/n) = ⁿ√a (the n-th root of a) - (ab)ⁿ = aⁿbⁿ

Surds: √a is called a surd when a is a positive rational number that is not a perfect square (or, for cube roots, not a perfect cube), making the root irrational.

Rules for surds: - √a × √b = √(ab) - √a ÷ √b = √(a/b) - Simplification: pull out perfect square factors, e.g., √72 = √(36×2) = 6√2 - √a + √b ≠ √(a+b) — this is the single most common error (see Section 11)

Rationalisation: To remove a surd from the denominator, multiply numerator and denominator by the conjugate.