Key Concepts: Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. Surds are irrational roots. Simplify surds by factoring out perfect squares.
Section A: Laws of Indices
1. Simplify using laws of indices: [4]
(a) 2³ × 2⁵
(b) (3²)³
(c) 5⁻²
(d) 161/2
2. Simplify: 8a6b32a2b5 [2]
Section B: Surds
3. Simplify the following surds: [4]
(a) √48
(b) √12 + √27
(c) 2√5 × 3√2
(d) √82
4. Rationalize the denominator: 3√5 [2]
Section C: Mixed Problems
6. Simplify: (√18 × √2) + (31/2)² [2]
Total marks: 20
Mark Scheme
1. (a) 2⁸ = 256 [1]
(b) 3⁶ = 729 [1]
(c) 1/25 [1]
(d) 4 [1]
2. (8÷2) × (a6a2) × (b3b5) = 4a4b−2 = 4a4b2 [2]
3. (a) √(16×3) = 4√3 [1]
(b) 2√3 + 3√3 = 5√3 [1]
(c) 6√10 [1]
(d) 2√2÷2 = √2 [1]
4. 3√5 × √5√5 = 3√55 [2]
5. 2(x-1) = 2³; x-1 = 3; x = 4 [2]
6. √36 + 3 = 6 + 3 = 9 [2]