Index Laws Objectives: A Grade Use Index Notation and Index Laws for Fractional Powers such as 16 A* Grade Use Index Notation and Index Laws for Fractional Powers such as 16 Prior knowledge: Understand : Use Index Laws for positive and negative powers

Index Laws Using the index laws to simplify a3 × a4 a3+4 = a7 Understanding that we add the index numbers we can deduce the meaning of: 100½ × 100½ By adding the index numbers 100½+½ = 1001 = 100 Therefore 100½ is a number, that, when multiplied by itself is 100. We know that: √100 × √100 = 100 Therefore: 100½ = √100 The general rule is: x½ = √x

The general rule is: x = a√x Index Laws Understanding that we add the index numbers we can deduce the meaning of: x × x × x By adding the index numbers 100 + + = 1001 = 100 3√100 × 3 √100 × 3 √100 = 100 Therefore: 100 = 3√100 The rule is: x = 3√x x = a√x The general rule is:

Now do these: 1. Evaluate a) 4½ b) 40 c) 8 d) 125 2. Evaluate a) 32 b) 243 c) 512 d) 59 049 Index Laws 2 1 2 5 2 3 8 9

Index Laws More complex fractional index laws together: (73)2 means (73) × (73) = 73+3 = 76 = 4 (4½)3 means 4½ × 4½ × 4½ 4 Therefore when we see We know it means (4½)3 In general a actually means (4 )y

Now do these: 1. Evaluate a) 4 b) 27 c) 16 d) 125 2. Evaluate a) 64 b) 243 c) 512 d) 6561 Index Laws 8 9 8 25 32 81 64 729

Now do these: 3. Evaluate a) -8 b) (-125) c) 4 d) 16 -0.5 e) 256 -0.25 f) 16 -0.25 g) 125 × 8 h) 49 × 81 i) 5-2 × 105 × 16 Index Laws 8 25 100 114 1000