Surds & Indices
Surds (Radicals)
- Definition: Irrational roots (√2, √3, ∛5, etc.)
- Pure surd: √a (only radical)
- Mixed surd: 2√3 (rational number × surd)
- Like surds: Same radical part (2√3, 5√3)
- Unlike surds: Different radical parts (√2, √3)
Surd Operations
- √a × √b = √(ab)
- √a / √b = √(a/b)
- (√a)² = a
- √a + √b ≠ √(a+b) [Important!]
- (√a + √b)(√a - √b) = a - b
Rationalization
- To rationalize 1/√a: Multiply by √a/√a = √a/a
- To rationalize 1/(a+√b): Multiply by (a-√b)/(a-√b)
- Conjugate of (a+√b) is (a-√b)
Laws of Indices (Exponents)
- a^m × a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- (a^m)^n = a^(mn)
- a^0 = 1 (a ≠ 0)
- a^(-m) = 1/a^m
- a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m
- (ab)^m = a^m × b^m
- (a/b)^m = a^m / b^m
Important Results
- a^(1/2) = √a
- a^(1/3) = ∛a
- If a^x = a^y, then x = y (when a ≠ 0, ±1)
- If a^x = b^x, then a = b (when x ≠ 0)
Quick Tips
- Always simplify surds to lowest form
- Use rationalization for division
- Remember: √a × √a = a
- For indices, add powers when multiplying same base