Algebra: Surds and Indices

Concept Explanation

Indices (exponents) follow laws of powers. Surds are irrational roots, usually square roots, simplified using root laws.

Index Laws

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m−n
• (a^m)ⁿ = a^mn
• a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ

Surd Laws

• √(ab) = √a × √b
• √(a/b) = √a / √b
• Rationalization: multiply numerator and denominator by the conjugate surd.

Example

1/(√5 − 2) = (√5 + 2) / [(√5)² − 2²] = √5 + 2.

Common Trap

√(a + b) ≠ √a + √b. Do not split sums under a root.