Algebra and Indices

ALGEBRA: Rules of Algebra and Indices. 1.1 Algebra Properties Let a, b, and c be real numbers, variables, or algebraic expressions. (a) Commutative Property of Addition: a + b = b + a Example: x + x2 = x2 + x Common Error x + x2 = x3 (???) (b) Commutative Property of Multiplication: ab = ba. Example: (3 – x) x3 = x3 (3 – x) (c) Associative Property of Addition: (a + b) + c = a + (b + c) Example: (x + 2) + x2 = x + (2 + x2) (3 + 2) + 42 = 3 + (2 + 42)

(f) Additive Identity Property: a + 0 = a. (d) Associative Property of Multiplication: (ab) c = a (bc) Example: (2x . 5)(3) = (2x) (5 . 3) (e) Distributive Properties: a(b + c) = ab + ac, (a + b)c = ac + bc. Examples: 2x(3 + 3x) = (2x . 3) + (2x . 3x) = 6x + 6x2, (y + 4)5 = (y . 5) + (4 . 5) = 5y + 20. (f) Additive Identity Property: a + 0 = a. (g) Multiplicative Identity Property: a . 1 = a. (h) Additive Inverse Property: a + (-a) = 0. (i) Multiplicative Inverse Property: a . 1/a = 1.

Remark: Because subtraction is defined as “adding the opposite”, the Distributive properties are also true for subtraction. For example, the “subtraction form” of a (b + c) = ab + ac is a(b –c) = ab – ac.

1.2 Properties of Negation Let a, b, and c be real numbers, variables, or algebraic expressions. (-1)a = -a. Example: (-1)6 = -6, (-1)5x = -5x. -(-a) = a. Example: -(-5) = 5, -(-y2) = y2. (-a)b = -(ab) = a(-b). Example: (-3)4 = -(4 . 3) = 4(-3). (-a)(-b) = ab. Example: (-2)(-x) = 2x. -(a + b) = (-a) + (-b) = – a – b. Example: -(x + 3) = (-x) + (-3) = – x – 3.

1.4 Properties of Zero Let a, b, and c be real numbers, variables, or algebraic expressions. 1. a + 0 = a and a – 0 = a. 2. a . 0 = 0. = 0, a ≠ 0. is undefined.

Note: There could be many possibilities: etc 5. Zero Factor Property: If ab = 0, then a = 0 or b = 0 or both = 0. Example: Common Error Note: There could be many possibilities: etc

Activity 1: Work in a group of 2 Activity 1: Work in a group of 2. Complete the following tasks in 15 minutes. Do it on a piece of paper. Task 1: Give an example of Associative Property of Subtraction Task 2: Give an example of Zero Factor Property. Task 3: Choose all the correct choices 1. 2. Task 4: Solve and .

1.5 Properties and Operations of Fractions Let a, b, c and d be real numbers, variables, or algebraic expressions such that b ≠ 0 and d ≠ 0. 1. Equivalent Fractions: if and only if . 2. Rules of Signs: and . 3. Generate equivalent Fractions: , c ≠ 0. (Eg. ½ = 2/4) 4. Add of Subtract with Like Denominators: . 5. Add of Subtract with Unlike Denominators: .

Multiply Fractions: . Divide Fractions: , c ≠ 0. 1.6 Indices 52 (‘5 squared’ or ‘5 to the power of 2’) and 43 (‘4 cubed’ or ‘4 to the power of 3’) are example of numbers in index form. 54 = 5×5×5×5, 31 = 3, 32 = 3×3, 43 = 4×4×4 etc. The 2, 3 and 4 are known as indices. Indices are useful (for example they allow us to represent numbers in standard form) and have a number of properties.

1.7 Laws of Indices: Multiplication: an × am = an+m. Eg. x2 × x5 = x7, 56 × 5-4 = 52 (because 6 + (-4)=6-4=2) x2 + x5 = x7 (???) Division: an ÷ am = an/am = an-m. Eg. x3/x5 = x -2 , 83 ÷ 8-2 = 83-(-2) = 85. 83 - 8-2 = 83-(-2) = 85 (???) Brackets: (an)m = anm. Eg. (x3)2 = x6, (42)4 = 48 (42)4 = 42+4 = 46 (???) Common Error Common Error Common Error

1.8 Further Index Properties Anything to power 0 is equal to 1, i.e., a0 = 1. Negative Indices: Eg. , , 1.9 Fractional Powers: x1/n means take the n-th root of x. Eg. 41/2 = , 641/3 = We can use the rule (an)m = an×m to simplify complicated index expressions. Eg. (1/8)-1/3 =[(1/8)-1]1/3 =[8]1/3 =[23]1/3 = 2.  

Exercises 1. Simplify: (i) (x3y5)4 x2y7÷x3y4 (iii) x3/y2 ÷ x6/y5 (x1/5y6/5 ÷ z2/5)5

2. Factorisation 2.1 Factors of a Number   If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number. So, the factors of 6 are 1, 2, 3 and 6. Example: Find all factors of 45. Solution: So, the factors of 45 are 1, 3, 5, 9, 15 and 45.

2.2 Common Factors Example: Find the common factor of 10 and 15. 10 = 2 × 5 = 1 × 10 Factors of 10: 1, 2, 5 and 10. 15 = 1 × 15 = 3 × 5 Factors of 15 : 1, 3, 5 and 15. Clearly, 5 is a factor of both 10 and 15.  It is said that 5 is a common factor of 10 and 15. Find a common factor of: a.  6 and 8 b.  14 and 21  

Solution: :

2.3 Highest Common Factor The highest common factor (HCF) of two numbers (or expressions) is the largest number (or expression) that is a factor of both. Consider the HCF of 16 and 24. The common factors are 2, 4 and 8.  So, the HCF is 8. Exe: Find the highest common factor of 60 and 150.

2.4 Highest Common Factor of Algebraic Expressions  The HCF of algebraic expressions is obtained in the same way as that of numbers. Example:

Solution:

2.5 Factorisation using the Common Factor We know that: a(b + c) = ab + ac The reverse process, ab + ac = a(b + c), is called taking out the common factor. Consider the factorisation of the expression 5x + 15.