Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Key Words and Introduction to Expressions 1 of 60 © Boardworks Ltd 2004 Writing expressions Here are some examples of algebraic expressions: 2 of 60 n+7 a number n plus 7 5–n 5 minus a number n 2n 2 lots of the number n or 2 × n 6 n 6 divided by a number n 4n + 5 4 lots of a number n plus 5 n3 a number n multiplied by itself twice or n×n×n 3 × (n + 4) or 3(n + 4) a number n plus 4 and then times 3. © Boardworks Ltd 2004 Writing expressions Miss Green is holding n number of cubes in her hand: Write an expression for the number of cubes in her hand if: She takes 3 cubes away. n–3 She doubles the number of cubes she is holding. 2n or 2×n 3 of 60 © Boardworks Ltd 2004 Equivalent expression match 4 of 60 © Boardworks Ltd 2004 Simplifying Algebraic Expressions 5 of 60 © Boardworks Ltd 2004 Like terms An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ). A term is made up of numbers and letter symbols but not operators. For example, 3a + 4b – a + 5 is an expression. 3a, 4b, a and 5 are terms in the expression. 3a and a are called like terms because they both contain a number and the letter symbol a. 6 of 60 © Boardworks Ltd 2004 Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 5+5+5+5=4×5 In algebra, a + a + a + a = 4a The a’s are like terms. We collect together like terms to simplify the expression. 7 of 60 © Boardworks Ltd 2004 Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, (7 × 4) + (3 × 4) = 10 × 4 In algebra, 7 × b + 3 × b = 10 × b or 7b + 3b = 10b 7b, 3b and 10b are like terms. They all contain a number and the letter b. 8 of 60 © Boardworks Ltd 2004 Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 2 + (6 × 2) – (3 × 2) = 4 × 2 In algebra, x + 6x – 3x = 4x x, 6x, 3x and 4x are like terms. They all contain a number and the letter x. 9 of 60 © Boardworks Ltd 2004 Collecting together like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting together like terms. An expression can contain different like terms. For example, 3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b = 7a + 8b This expression cannot be simplified any further. 10 of 60 © Boardworks Ltd 2004 Collecting together like terms Simplify these expressions by collecting together like terms. 1) a + a + a + a + a = 5a 2) 5b – 4b = b 3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6 = 2c + 2d + 9 4) 4n + n2 – 3n = 4n – 3n + n2 = n + n2 5) 4r + 6s – t 11 of 60 Cannot be simplified © Boardworks Ltd 2004 Algebraic perimeters Remember, to find the perimeter of a shape we add together the lengths of each of its sides. Write algebraic expressions for the perimeters of the following shapes: 2a Perimeter = 2a + 3b + 2a + 3b 3b = 4a + 6b 5x 4y x 5x 12 of 60 Perimeter = 4y + 5x + x + 5x = 4y + 11x © Boardworks Ltd 2004 Using Index Notation 13 of 60 © Boardworks Ltd 2004 Using index notation Simplify: x + x + x + x + x = 5x Simplify: x to the power of 5 x × x × x × x × x = x5 This is called index notation. Similarly, x × x= x2 x × x × x = x3 x × x × x × x = x4 14 of 60 © Boardworks Ltd 2004 Using index notation We can use index notation to simplify expressions. For example, 3p × 2p =3 × p × 2 × p = 6p2 q2 × q3 = q × q × q × q × q = q5 3r × r2 =3 × r × r × r = 3r3 2t × 2t =(2t)2 15 of 60 or 4t2 © Boardworks Ltd 2004 Expressions of the form (xm)n When a term is raised to a power and the result raised to another power, the powers are multiplied. For example, (a5)3 = a5 × a5 × a5 = a(5 + 5 + 5) = a15 = a(3 × 5) In general, (xm)n = xmn 16 of 60 © Boardworks Ltd 2004 Expressions of the form (xm)n Rewrite the following without brackets. 1) (2a2)3 = 8a6 2) (m3n)4 = m12n4 3) (t–4)2 = t–8 4) (3g5)3 = 27g15 5) (ab–2)–2 = a–2b4 6) (p2q–5)–1 = p–2q5 7) (h½)2 = h 8) (7a4b–3)0 = 1 17 of 60 © Boardworks Ltd 2004 The zero index Look at the following division: Any number or term divided by itself is equal to 1. 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, for all x 0, x0 = 1 18 of 60 © Boardworks Ltd 2004 Index laws Here is a summary of the index laws. xm × xn = x(m + n) (xm)n = xmn x0 = 1 (for x = 0) x1 = x x = x 1 2 19 of 60 © Boardworks Ltd 2004 Multiplying Algebraic Terms 20 of 60 © Boardworks Ltd 2004 Multiplying terms together In algebra we usually leave out the multiplication sign ×. Any numbers must be written at the front and all letters should be written in alphabetical order. For example, 4 × a =4a 1 × b =b We don’t need to write a 1 in front of the letter. b × 5 =5b We don’t write b5. 3 × d × c = 3cd We write letters in alphabetical order. 6 × e × e = 6e2 21 of 60 © Boardworks Ltd 2004 Brackets Look at this algebraic expression: 4(a + b) What do you think it means? Remember, in algebra we do not write the multiplication sign, ×. This expression actually means: 4 × (a + b) or (a + b) + (a + b) + (a + b) + (a + b) =a+b+a+b+a+b+a+b = 4a + 4b 22 of 60 © Boardworks Ltd 2004 Expanding Algebraic Expressions 23 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example, 2(5 – x) We need to multiply the bracket by 2. 10 – 2x 24 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 4 – (5n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 – 5n + 3 = 4 + 3 – 5n = 7 – 5n 25 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 2(3n – 4) + 3(3n + 5) We need to multiply out both brackets and collect together like terms. 6n – 8 + 9n + 15 = 6n + 9n – 8 + 15 = 15n + 7 26 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 5(3a + 2b) – 2(2a + 5b) We need to multiply out both brackets and collect together like terms. 15a + 10b – 4a –10b = 15a – 4a + 10b – 10b = 11a 27 of 60 © Boardworks Ltd 2004