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AS-Level Maths: Core 1 for Edexcel C1.1 Algebra and functions 1 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. 1 of 38 © Boardworks Ltd 2005 Using and manipulating surds Using and manipulating surds Contents Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 2 of 38 © Boardworks Ltd 2005 Types of number We can classify numbers into the following sets: The set of natural numbers, : Positive whole numbers {0, 1, 2, 3, 4 …} The set of integers, : Positive and negative whole numbers {0, ±1, ±2, ±3 …} The set of rational numbers, : Numbers that can be expressed in the form mn , where n and m are integers. All fractions and all terminating and recurring decimals are rational numbers; for example, ¾, –0.63, 0.2. The set of real numbers, : All numbers including irrational numbers; that is, numbers that cannot be expressed in the form mn , where n and m are integers. For example, and 2 . 3 of 38 © Boardworks Ltd 2005 Introducing surds 2, √3 3 or 5, When the square root of a number, for example √2, or √5,is irrational, it is often preferable to write it with the root sign. Numbers written in this form are called surds. Can you explain why √1.69 1.69 is not a surd? 1.69 is not a surd because it is not irrational. √1.69 169 1.69 = 100 169 = 100 13 = 10 = 1.3 4 of 38 This uses the fact that a a = b b © Boardworks Ltd 2005 Manipulating surds When working with surds it is important to remember the following two rules: ab = a × b and a a = b b Also: a× a =a You should also remember that, by definition, √aa means the positive square root of a. 5 of 38 © Boardworks Ltd 2005 Simplifying surds We are often required to simplify surds by writing them in the form a b . We can do this using the fact that For example: ab = a × b . Simplify 50 by writing it in the form a b . Start by finding the largest square number that divides into 50. This is 25. We can use this to write: 50 = 25× 2 = 25 × 2 =5 2 6 of 38 © Boardworks Ltd 2005 Simplifying surds Simplify the following surds by writing b. them in the form aa√b. 45 = 9×5 7 of 38 3) 3 40 2) 98 1) 45 98 = 49× 2 3 40 = 3 8 × 5 = 9× 5 = 49 × 2 = 3 8×3 5 =3 5 =7 2 = 23 5 © Boardworks Ltd 2005 Simplifying surds 8 of 38 © Boardworks Ltd 2005 Adding and subtracting surds Surds can be added or subtracted if the number under the square root sign is the same. For example: Simplify 45 + 80. Start by writing 45 and 80 in their simplest forms. 45 = 9×5 80 = 16×5 = 9× 5 = 16 × 5 =3 5 =4 5 45 + 80 = 3 5 + 4 5 = 7 5 9 of 38 © Boardworks Ltd 2005 Expanding brackets containing surds Simplify the following: 1) (4 2)(1+ 3 2) 2) ( 7 2)( 7 + 2) = 4 +12 2 2 6 = 7+ 2 7 2 7 2 = 11 2 2 =5 Problem 2) demonstrates the fact that (a – b)(a + b) = a2 – b2. In general: ( a b )( a + b ) a b 10 of 38 © Boardworks Ltd 2005 Rationalizing the denominator Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 11 of 38 © Boardworks Ltd 2005 Rationalizing the denominator When a fraction contains a surd as the denominator we usually rewrite it so that the denominator is a rational number. This is called rationalizing the denominator. For example: 5 Simplify the fraction . 2 In this example we rationalize the denominator by multiplying the numerator and the denominator by 2. × 2 5 5 2 = 2 2 × 2 12 of 38 © Boardworks Ltd 2005 Rationalizing the denominator Simplify the following fractions by rationalizing their denominators. 2 1) 3 2) × 3 2 2 3 = 3 3 × 3 13 of 38 2 5 3 3) 4 7 × 5 2 = 5 × 5 × 7 10 5 3 3 7 = 28 4 7 × 7 © Boardworks Ltd 2005 Rationalizing the denominator When the denominator involves sums of differences between surds we can use the fact that (a – b)(a + b) = a2 – b2 to rationalize the denominator. For example: Simplify 1 5 2 = 1 . 5 2 1 5 2 × 5 +2 5 +2 5 +2 = 54 = 5 +2 14 of 38 © Boardworks Ltd 2005 Rationalizing the denominator More difficult examples may include surds in both the numerator and the denominator. For example: 2 3 1 . Simplify 3 +1 2 3 1 (2 3 1)( 3 1) = 3 +1 ( 3 +1)( 3 1) 73 3 = 3 1 73 3 = 2 15 of 38 Working: 2 3 1 3 1 = 6 2 3 3 + 1 =7 3 3 © Boardworks Ltd 2005 The index laws Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 16 of 38 © Boardworks Ltd 2005 Index notation Simplify: a to the power of 5 a × a × a × a × a = a5 a5 has been written using index notation. The number a is called the base. an The number n is called the index, power or exponent. In general: n of these an = a × a × a × … × a 17 of 38 © Boardworks Ltd 2005 Index notation Evaluate the following: 0.62 = 0.6 × 0.6 = 0.36 34 = 3 × 3 × 3 × 3 = 81 (–5)3 = –5 × –5 × –5 = –125 When we raise a negative number to an odd power the answer is negative. 27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4 × –4 × –4 × –4 = 256 18 of 38 When we raise a negative number to an even power the answer is positive. © Boardworks Ltd 2005 The multiplication rule When we multiply two terms with the same base the indices are added. For example: a4 × a2 = (a × a × a × a) × (a × a) =a×a×a×a×a×a = a6 = a (4 + 2) In general: am × an = a(m + n) 19 of 38 © Boardworks Ltd 2005 The division rule When we divide two terms with the same base the indices are subtracted. For example: a5 ÷ a2 a×a×a×a×a = a×a = a3 = a (5 – 2) 2 4p6 ÷ 2p4 4×p×p×p×p×p×p = = 2p2 = 2p(6 – 4) 2×p×p×p×p In general: am ÷ an = a(m – n) 20 of 38 © Boardworks Ltd 2005 The power rule When a term is raised to a power and the result raised to another power, the powers are multiplied. For example: (y3)2 = y3 × y3 (pq2)4 = pq2 × pq2 × pq2 × pq2 = (y × y × y) × (y × y × y) = p4 × q (2 + 2 + 2 + 2) = y6 = y3×2 = p4 × q8 = p4q8 = p1×4q2×4 In general: (am)n = amn 21 of 38 © Boardworks Ltd 2005 Using index laws 22 of 38 © Boardworks Ltd 2005 Zero and negative indices Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 23 of 38 © Boardworks Ltd 2005 The zero index Any number or term divided by itself is equal to 1. Look at the following division: y4 ÷ y4 = 1 But using the rule that xm ÷ xn = x(m – n) y4 ÷ y4 = y(4 – 4) = y0 That means that y0 = 1 In general: a0 = 1 24 of 38 (for all a ≠ 0) © Boardworks Ltd 2005 Negative indices Look at the following division: b2 ÷ b4 = b×b b×b×b×b 1 1 = = 2 b×b b But using the rule that am ÷ an = a(m – n) b2 ÷ b4 = b(2 – 4) = b–2 That means that b–2 = 1 b2 In general: a–n = 25 of 38 1 an © Boardworks Ltd 2005 Negative indices Write the following using fraction notation: 26 of 38 1 u This is the reciprocal of u. 1) u–1 = 2) 2n–4 = 2 n4 3) x2y–3 = x2 y3 4) 5a(3 – b)–2 = 5a (3 b)2 © Boardworks Ltd 2005 Negative indices Write the following using negative indices: 2 1) = 2a–1 a x3 2) 4 = x3y–4 y p2 3) = p2(q + 2)–1 q+2 3m 4) 2 = 3m(n2 – 5)–3 3 (n 5) 27 of 38 © Boardworks Ltd 2005 Fractional indices Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 28 of 38 © Boardworks Ltd 2005 Fractional indices Indices can also be fractional. For example: 1 2 What is the meaning of a ? Using the multiplication rule: 1 2 1 2 1+1 2 2 a × a =a = a1 =a a= a × a But 1 2 So a = a 29 of 38 1 2 a is the square root of a. © Boardworks Ltd 2005 Fractional indices Similarly: 1 3 1 3 1 3 a ×a ×a = a 1+1 +1 3 3 3 = a1 =a a = 3 a × 3 a ×3 a But 1 3 1 3 So a =3a a is the cube root of a. In general: 1 n a =na 30 of 38 © Boardworks Ltd 2005 Fractional indices 2 3 What is the meaning of a ? 2 3 We can write a as a 2 31 . Using the rule that (am)n = amn, we can write 2 3 1 3 a (a ) 3 a 2 2 3 We can also write a as a 1 2 3 2 3 2 . 1 3 a (a )2 ( 3 a )2 In general: m n n a = a 31 of 38 m or m n a = a n m © Boardworks Ltd 2005 Fractional indices Evaluate the following: 1) 16 5 4 2) (0.125) 5 4 4 5 16 = ( 16) 32 3) 36 32 (0.125) = 8 2 3 36 32 32 = 1 ( 36 )3 = 25 = ( 3 8)2 = 32 = 22 1 = 3 6 =4 = 32 of 38 1 216 © Boardworks Ltd 2005 Summary of the index laws Here is a summary of the index laws for all rational exponents: a m × a n = a ( m+n ) m n a ÷a =a m n (a ) = a ( m n ) mn 1 a =a a = 1 (for a 0) 0 33 of 38 a n 1 = n a 1 2 a = a 1 n n a = a m n n a = a m © Boardworks Ltd 2005 Solving equations involving indices Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 34 of 38 © Boardworks Ltd 2005 Solving equations involving indices We can use the index laws to solve certain types of equation involving indices. For example: Solve the equation 25x = 1255 – x. 25x = 1255 – x (52)x = (53)5 – x 52x = 53(5 – x) 2x = 3(5 – x) 2x = 15 – 3x 5x = 15 x=3 35 of 38 © Boardworks Ltd 2005 Examination-style questions Contents Using and manipulating surds Rationalizing the denominator The index laws Zero and negative indices Fractional indices Solving equations involving indices Examination-style questions 36 of 38 © Boardworks Ltd 2005 Examination-style question 1 6+ 3 Show that can be written in the form a + b 2 where 6 3 a and b are integers. Hence find the values of a and b. Multiplying top and bottom by 6 + 3 gives 3 6+ 3 6 6+ So 37 of 38 = 6+2 6 3 +3 63 3 6+ 3 9 + 2 18 18 can be written as 3 2. = 3 9+6 2 = 3 =3+2 2 a = 3 and b = 2 © Boardworks Ltd 2005 Examination-style question 2 a) Express 32x in the form 2ax where a is an integer to be determined. b) Use your answer to part a) to solve the equation x 32 = 2 x2 32 = 25 So 32x = (25)x Using the rule that (am)n = amn 32x = 25x b) Using the answer from part a) this equation can be written as 5x x2 2 =2 5x = x2 5x – x2 = 0 x (5 – x) = 0 x = 0 or x = 5 a) 38 of 38 © Boardworks Ltd 2005