Starter Estimate the answer to the following: √27 √105 8.92

Index Laws Learning Objectives: Level 7/Grade B 29/04/2019 Learning Objectives: Able to estimate the answer to a question involving indices Able to multiply terms with the same base Able to divide terms with the same base

Index Laws How can I simplify: 45 x 43? How can I simplify: 79 x 76? 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 48 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 715

Law 1 : Multiplication 26 x 24 = 210 24 x 22 = 26 35 x 37 = 312 General Rule: When multiplying terms with the same base, you add the powers am x an = am+n EXAMPLE: 47 x 45 = 47+5 = 412

Index Laws How can I simplify 95 ÷ 93? 9 x 9 x 9 x 9 x 9 = 9 x 9 = 92 11 x 11 x 11 x 11 x 11 x 11 x 11 = 115 11 x 11

Law 2 : Division 26 ÷ 24 = 22 25 ÷ 22 = 23 35 ÷ 37 = 3-2 General Rule When dividing terms with the same base, you subtract the powers am ÷ an = am-n EXAMPLE: 47 ÷ 45 = 47-5 = 42

Index Laws How can I simplify (43)4? How can I simplify (72)5? 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 412 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 710

Law 3 : Brackets (26)2 = 26 x 26 = 212 (35)3 = 35 x 35 x 35 = 315 General Rule: When raising a power to another power, you multiply the powers (am)n = am x n EXAMPLE: (47)5= 47 x 5 = 435

Index Laws Simplify the following: 73 x 78 75 𝑥4 x 𝑥9 𝑥5 x 𝑥10

Index Laws Crack the code to reveal an hilarious joke!!

(a) 32 x 37 (b) 79 x 711 (c) y2 x y5 (d) p5 x p6 x p3 (e) 99 ÷ 92 (f) 417 ÷ 412 (g) h6 ÷ h4 (h) g15 ÷ g12 ÷ g2 (j) 35 ÷ 37 (k) 53 x 5-5 (l) x-3 ÷ x-6 (m) k-7 x k12 (n) a15 x a-2 x a-7 (o) e-3 ÷ e-12 ÷ e6 (p) y5 x y-7 ÷ y3 (q) d-4 ÷ d9 x d-4 (r) (s3)12 (s) (g-2)6 (t) (w5)-3 (u) (b-6)-4 (v) 5t2 x 7t5 (w) 6c5 ÷ 2c3 (x) (9g4)2 (y) (10a2)3

Index Laws Learning Objectives: Level 8 29/04/2019 Learning Objectives: Able to multiply terms with the same base Able to divide terms with the same base Able to simplify terms with negative indices

Index Laws Complete the Tarsia puzzle by matching the correct simplified indices to the questions

Index Laws How can I simplify 5a3 x 6a7? 5 x a x a x a x 6 x a x a x a x a x a x a x a 5 x 6 x a x a x a x a x a x a x a x a x a x a = 30a10

Algebra and Indices 9d6 x 8d3 12e10 x 6e4 24f8 ÷ 6f4 60g12 ÷ 5g3

Starter 89 ÷ 83 (35)8 4𝑥7 x 7𝑥6 (2y3)4

Index Laws Learning Objectives: Level 8 29/04/2019 Learning Objectives: Able to multiply and divide terms with the same base Able to raise a power to another power Able to simplify terms with negative indices

Algebra and Indices (7c3)2 = 7 x c x c x c x 7 x c x c x c = 7 x 7 x c x c x c x c x c x c = 49c6 (4j8)3 (5k-3)4 7 x c x c x c x 7 x c x c x c

Law 4: Negative Indices General Rule a-n = 1 an 25 = 32 24 = 16 23 = 8 22 = 4 21 = 2 General Rule a-n = 1 an 20 = 1 2-1 = ½ 2-2 = ¼ 2-3 = ⅛ 2-4 = 20 = 2-1 = 2-2 = 2-3 = 2-4 = 1 16

Law 5: Power 0 25 = 32 24 = 16 23 = 8 22 = 4 GENERAL RULE: 21 = 2 Any term to the power 0 is equal to 1. 20 = 1 2-1 = ½ 2-2 = ¼ 2-3 = ⅛ 2-4 = 1 16

Negative Indices = 1 6 2 = 1 36 = 1 7 3 = 1 343 = 9 x 1 𝑚 4 = 9 𝑚 4 = 1 6 2 = 1 36 6-2 7-3 9m-4 = 1 7 3 = 1 343 = 9 x 1 𝑚 4 = 9 𝑚 4

Simplify the following, leaving your answers in index notation. 1. (38)4 = 332 2. 7f3 x 9f9 = 63f12 3. 124 x 1210 = 1214 4. (6m5)3 = 216m15 5. 25 ÷ 2-3 = 28 6. 42g7 ÷ 7g2 = 6g5 7. 39 x 36 8. 116 x 116 = 310 = 117 35 112 x 113 Solve the following. = 1 49 = 1 27 9. 7-2 10. 3-3

Negative Indices ANSWERS: 1 9 6 b) 1 11 5 c) 1 24 19 d) 1 64 e) 1 16 f) 1 125 g) 10 𝑔 5 h) 3 ℎ 8 i) 13 𝑖 10 Simplify: 9 −6 b) 11 −5 c) 24 −19 Solve: d) 8 −2 e) 2 −4 f) 5 −3 g) 10𝑔 −5 h) 3ℎ −8 i) 13𝑖 −10

More difficult examples 46 6p8 x 3p3 9p4 x p7 Simplify: Solve: 83 85 (4-1)3

Some practice… Exercise 10C, page 214 Exercise 10B, page 212 Q1, 2 Q4, 5, 6 (a to c for each) Q7, 8 Exercise 10B, page 212