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Per Year 10A Module 1 Indices Lesson 1: Index Laws Homework: 3F (p. 186): # 4, 6, 8, 11 ; 3G (p. 186): # 1-3, 5, 9, 10 Lesson 2: Rational (Fractional) Indices Materials Essential mathematics 3.6 - 3.9 Homework: 3I (p. 200): # 5, 8-10 Lesson 3: Scientific Notation Homework: 3H (p. 195): # 4-6 Time Required 3 periods Test date:________________l Indices 1 Part 1: Indices Laws: Things to Know: Law 1: x a x b x a b Describe in words: How it’s done: Simplify 42 × 45 giving your answer in index form. 42 × 45 = 42 + 5 When multiplying terms with the same base, you add the indices. The bases do not change. 7 = 4 Simplify 3k2 × 4k8. 3k2 × 4k8 = 12k10 When multiplying pronumerals (or variables) with coefficients: Calculate the product of the coefficients. Check that the bases are the same. If the bases are the same, add the indices. You try: Indices 2 Things to Know: Law 2: x a xb x a b Describe in words: How it’s done: Simplify 612 ÷ 63. 612 ÷ 63 = 6 12 – 3 = 69 Simplify 15p10 ÷ 3p. In divisions with terms having the same base, subtract the index of the divisor (second index). Be careful: This only works when the bases are the same. Remember that when the index is 1 it can be omitted. 15p10 ÷ 3p = 5p10 ÷ p1 = 5p10 – 1 = 5p9 You try: Simplify: You try: Simplify: Indices 3 Things to Know: Law 3: x 0 1 Describe in words: How it’s done: Calculate the value of each of these expressions. 1. y0 = 1 2. 5y0= 5 × 1= 5 3. (5y)0 = 1 4. y0 + 70= 1 + 1= 2 The order of operation is very important in all calculations: brackets powers multiplication & division addition & subtraction Any number to the power of zero = 1 (Remember the debate about 00 which might be undefined or might be 1 depending on how you look at it.) You try: Simplify: Indices 4 Things to Know: Law 4: ( x a )b x ab Describe in words: How it’s done: Write (73)2 as an equivalent expression in simple index form. (73)2 = 7 3×2 = 7 6 Expand (h5)4. (h5)4 = h5 × 4 =h9 Expand (3y7)3. (3y7)3 = 33 × y7 × 3 = 27y21 When raising an expression in index form to another power, multiply the indices and leave the base unchanged. Wherever possible, determine a method to check your solutions. Raise both factors inside the brackets to the power 3: the coefficient, 3, and the power of the pronumeral, y7. 33 = 27 = (y7)3 = y7 × 3 When raising a base and power to another power, multiply the indices. Leave the base unchanged. Another method is to write out the full multiplication. You try: Expand and simplify: Indices 5 Things to Know: Law 5: x a Describe in words: 1 xa How it’ done: What number does 4–2 represent? 4–2 is the reciprocal of 42. 42 = 16. 4–2 = So 4–2 = = The negative index applies to the pronumeral p only. Express 4p–3 as a fraction. 4p–3 = 4 × p–3 = 4 × = p–3 = You try: Simplify, leaving answer with a positive index: Indices 6 A more complex example: Simplify ÷ 6m4. Express your answer in index form without fraction notation. ÷ 6m4= 24m–5 ÷ 6m4= 4m-5 -4= 4m- 9 Converting an expression from fractions to index form can help you simplify calculations using index laws. Note that this answer expressed in fraction form would be . You try: Indices 7 Simplify: Lesson 1 Homework: 3F (p. 186): # 4, 6, 8, 11 ; 3G (p. 191): # 1-3, 5, 9, 10 Indices 8 Part 2: Rational (Fractional) indices: Things to Know: The same laws that apply to natural number indices, apply to rational (fractional) indices How it’s done: Express in root form then as a basic numeral. is the fifth root of 32. A base raised to the power base. is the mth root of that = So Simplify ( )8= ( = Taking roots and raising to powers are inverse operations, just like ÷ and × are inverse operations. )8. ×8 = k4 When raising a power to a further power, multiply the indices. (am)n = a m × n = amn You can also apply this to roots and powers. = So ( )8= ( )8 = k4 Simplify p × p × q ÷ q . When simplifying an expression containing several pronumerals, deal with each pronumeral separately. The index laws apply to all types of indices: p ×p ×q÷q = p1 × q ÷ q when multiplying terms with the same base, add the indices when dividing terms with the same base, subtract the second index from the first index =p×q÷q =p×q = pq You try: Indices 9 Evaluate: Use your understanding of index laws and indices to evaluate By writing roots in index form, you can evaluate this expression without the need for a calculator )–4. ( = )–4 = ( )–4 ( = 5–2 = = You Try: Lesson 2 Homework: 3I (p. 200): # 5, 8-10 Indices 10 Part 3: Scientific notation Things to Know: 1. To express a large number in scientific notation, write it in the form: (NUMBER BETWEEN 1 AND 10) × 10POWER 2. To express a small number in scientific notation, write in form (NUMBER BETWEEN 1 AND 10) × 10 –POWER How it’s done: 4×2=8 Calculate 4 × 10–3 × 2 × 108. 10–3 × 108 = 105 5 5 66826111 0 2 You could also write both numbers as basic numerals before calculating. 5 5 4 × 10–3=4 × Express your answer in scientific notation. 2 = –3 8 4 × 10 × 2 × 10 = 8 × 10 5 2 × 108 =200 000 000 You try: Write in Scientific notation: 3. Calculate, leaving answer in scientific notation: Indices 11 Lesson 3 Homework: 3H (p. 195): # 4-6 , 7 Indices 12