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Prime Factors/ Indices Objectives By the end of the lesson learners will be able to: • Identify Prime Factors (Process of the Factor Tree) • Express a number as a product of its prime factors. • Identify the LCM and LCF using Venn diagrams. • Use powers (also known as indices) Prime Factors • We can use a ‘factor tree’ to enable us to write a number as “a product of its prime factors” • Each time you reach a prime number, you stop and circle that number • Example: Write 84 as a product of its prime factors 84 2 42 2 • 84 = 2 x 2 x 3 x 7 • 84 = 22 x 3 x 7 21 3 7 Prime Factors • Write 420 as a product of prime factors. Your turn… 1. Find the HCF of the following: 2. Find the LCM of the following: a) b) c) d) e) f) a) b) c) d) e) f) 18 and 28 16 and 40 42 and 90 40 and 63 20, 64 and 108 54, 90 and 162 2 8 6 1 4 18 4 and 5 8 and 12 6 and 9 12 and 15 5, 8 and 10 4, 7 and 9 20 24 18 60 40 252 3. Write the following as products of their prime factors. 2 x 7 x 11 32 x 2 33 x 5 2 x 3 x 5 x 7 x 11 a) 18 b) 135 c) 154 d) 2310 HCF and LCM • Calculate the HCF and LCM of 48 and 60 48 48 = 2 x 2 x 2 x 2 x 3 60 48 = 2 x 2 x 2 x 2 x 3 60 = 2 x 2 x 3 x 5 HCF = 2 x 2 x 3 = 12 LCM = 2 x 2 x 3 x 2 x 2 x 5 = 240 Find the HCF and LCM of the following: 24 and 30 36 and 50 48 60 16 and 72 • Multiply the numbers in the overlap to get the HCF • Multiply all the numbers in the Venn diagram to get the LCM. 3 5 5 8 Introduction 2 What does 4 actually mean? 4×4 16 Introduction 5 What does 4 actually mean? 4×4×4×4×4 Introduction How can we write 5×5×5×5 5 4 This is 5×5×5×5 written in INDEX FORM Writing in Index Form Write the following in Index Form • 8×8×8×8 = 84 6 • 3×3×3×3×3×3 = 3 • 9×9×9 = 93 Indices • 3x3x3x3 =3 4 • 3x3x3 =3 3 • 3x3 =3 2 • 3 =3 1 Multiplying Indices Simplify the following leaving in index form. 3 4 3 4 • 88 × 8 8 = 8×8×8 × 8×8×8×8 =8 7 Is there a quick way you can work this out? • 3 4 8 ×8 3 = 4 8 + =8 7 Remember this only works for numbers with the Same base number Multiplying Indices Simplify the following leaving in index form. 2 4 2 4 • 55 × 5 5 = 5×5 × 5×5×5×5 =5 6 The Quick Way!!! • 2 4 5 ×5 2 = 4 5 + =5 6 Remember this only works for numbers with the Same base number Dividing Indices Simplify the following leaving in index form. 5 2 5 2 x 2 x 2 x 2 x 25 1. 22 ÷ 2 2=2 = 2 x 2 22 =2 3 The Quick Way!!! 1. 5 2 2 ÷2 5 = 2 2 - =2 3 Remember this only works for numbers with the Same base number 2 Indices Simplify the following leaving in index form. 3 3 3 1. (( 2 ) 2 2 x2 This means this =2 6 For these it is best to write them out! Remember this only works for numbers with the Same base number Indices Simplify the following leaving in index form. 5 5 5 5 1. (( 3 ) 3 3 x3 x3 This means this =3 For these it is best to write them out! Remember this only works for numbers with the Same base number 15 Questions Simplify the following numbers, leaving your answers in index form: 1. (a) 23 x 22 25 (b) 35 x 32 37 (c) 73 x 7 74 (d) 95 x 92 97 2. (a) 23 ÷ 22 (b) 35 ÷ 32 (c) 73 ÷ 7 (d) 95 ÷ 92 2 33 72 93 Indices with Letters • axaxaxa • axaxa • axa =a 4 =a 3 =a 2 =a 1 • a Multiplying Indices Simplify the following leaving in index form. 3 4 3 4 • aa × a a = × a×a×a×a a×a×a =a 7 Is there a quick way you can work this out? • 3 4 a ×a 3 = 4 a + =a 7 Remember this only works for numbers with the Same base number Indices with Letters a55 x 2a2 33a = 3xaxaxaxaxa = 3 x 2x = 6a7 a x 2xaxa x a x a x a x a x a x a Indices with Letters 66aa33 x 33aa22 = 6 x a x a x a x = 6 x 3 x a x a x a = 18a5 3 x a x a x The Quick Way!!! 66aa33 x 33aa222 = x x a + = 18a5 axa Questions Simplify the following numbers, leaving your answers in index form: 1. (a) a3 x a2 (b) a5 2. (a) a5 x a6 (c) a x a3 a11 (23)2 26 3 (b) 5 3. (a) 3b x 7b 8 21b a2 x a12 a4 (35)2 310 (c) 7 (b) 5b x 2b 10b8 (d) a (73)3 79 3 (d) 3 (c) 2b x 6b 12b6 14 (92)5 10 9 3 2 (d) 2b x 15b 5 30b