DEPARTMENT OF MATHEMATICS Core 1 Topic 1 and Topic 2 – Surds and Indices 1 Surds Multiplication and Division of Surds Simplifying Surds Rationalising the denominator Write a single surd in simplest form Rationalise the denominator with a single surd Multiply surds with integers Write an expression with multiple surds in its simplest form Rationalise the denominator with a surd of the form a ± √b Expand Brackets with surds Simplify a fractional surd with an integer denominator Rationalise the denominator with a surd of the form √a ± √b Multiply two surds together Divide two surds Simplify expressions containing a mix of the above 2 Surds A little introduction: The Natural Numbers, represented by the letter ℕ, are the positive whole numbers {1, 2, 3, 4, …}. They are sometimes called the ‘counting numbers’. The Integers, ℤ, include all of the whole numbers {…, -3, -2, -1, 0, 1, 2, 3, 4, …} The Rational Numbers, ℚ, are all numbers that can be written as ⁄ , where and are integers and ≠ 0. Basically – all positive and negative fractions with integer numerators and denominators! The Irrational Numbers (they don’t have a letter!) are numbers that cannot be written as a fraction in the form above. Basically – numbers that go on forever with no repeats or patterns (think , , √2, etc) The Real Numbers, ℝ, are all numbers on a number line (i.e. every number you can possibly think of!) The Complex Numbers, ℂ, are not studied in A Level Maths, but they are pretty cool nonetheless so we’ll briefly look at them. They involve something called the imaginary number, , that is equal to √−1. Surds are roots that cannot be expressed as rational numbers. If you type a surd into your calculator and press ‘equals’, then the calculator will round that surd to around 9 decimal places. In reality, that number continues for ever, and without any repeats. You should always write a surd in its exact form (i.e. as a √ ), unless you are asked to round your answer. Rules for manipulating surds: √ × √ = √ √ √ =√ √ × √ = √2 = 3 We can simplify surds using these rules. Worked Examples: a) √20 b) √72 Simplify: 4 c) √28 2 d) √150 − √96 + √24 5 Notes: 6 Rationalising the Denominator: Rationalising the denominator is the process of removing surds from the denominator of a fraction. There are two different types of question that we will be expected to rationalise. Let’s study the first type: Worked Examples: a) b) Rationalise: 1 √3 5√2 2√6 7 And now, let’s study the second type of question that we can be expected to rationalise: Worked Examples: a) b) Rationalise: 1 2+√3 2 1−√5 8 c) √5 √3+√2 9 Notes: 10 Simplifying an expression containing surds: We can combine these different skills to simplify an expression that contains surds. Let’s look at some examples of this: Worked Examples: a) 5+√2 b) √80 + Simplify fully the following expressions: √3−1 30 √5 11 c) (1+√3)(2−√2) √6 12 Surds – Questions Exercise 1: Exercise 2: 13 Exercise 3: 14 Solutions to all exercises: Exercise 1: Exercise 2: 15 Exercise 3: 16 Indices Basic simplification of indices Simplifying fractional Indices Apply the rule: am x an = am+n Apply the rule: = − Simplification of negative indices Apply the rule: 1 = Apply the rule: = Advanced Indices Apply the rule: 1 − = Simplifying algebraic expressions involving indices Apply the rule: − = Solve an equation by changing bases Apply the rule: = 17 Indices Key terms: → x is the base n is the index/power When we are working with numbers that have the same base, we can apply the following rules to multiply and divide the numbers: Multiplication: × = + ‘When multiplying, add the indices’ Division: = − ‘When dividing, subtract the indices’ Raising a power to a power: ( ) = ‘When raising to a power, multiply the indices’ Worked Examples: Simplify: a) 5 × 2 e) (4 )2 × 3 b) 5 3 × 3 × 4 f) (22 )3 ÷ 125 c) d) 7 3 9 5 ÷ 3 −2 1 2 g) × 3 h) 23 × 42 1 3 18 Negative and Fractional Indices: Negative Indices: − = 1 and − ( ) =( ) ‘The ‘flipped’ number is called the reciprocal’ Fractional Indices (Basic): 1 = √ ‘In a fractional index, the denominator is the root’ Fractional Indices (Advanced): = √ ‘In a fractional index, the denominator is the root, and the numerator is the power. The order in which we do these doesn’t matter – pick the easiest!’ 19 Worked Examples: a) Evaluate the following: 5−2 2 −3 b) ( ) c) 49 1 2 d) 1 3 3 8 20 3 2 e) 25 f) 49−2 1 1 g) h) 49 ( ) 2 81 ( 8 ) − 2 3 125 21 Notes: 22 Converting an expression into the form : Later on in C1, we will study both differentiation and integration. In these topics, it is crucial that we can convert an expression into the form . Worked Examples: a) b) Write in the form : 1 3 3√ 2 23 c) d) 2 3 5 √ 1 2 4 24 Notes: 25 Simplifying more complex expressions: We also need to be able to simplify expressions with more than one letter. The rules studied so far still apply: Worked Examples: Simplify: a) 52 3 × 2 4 b) 354 3 ÷ −72 5 c) √9 4 (3 2 )2 26 Notes: 27 Solving equations with indices: Finally, we are expected to be able to solve simple equations involving indices. It is important to check that the bases are the same when you wish to apply the rules in an equation with multiple terms! Worked Examples: a) 1/2 = 7 b) 3 3 = 81 Solve: 3 2 c) =8 d) 32+1 = 92−1 28 Notes: 29 Indices – Questions Exercise 1: 30 Exercise 2: Exercise 3: 31 Exercise 4: Exercise 5: (Calculator Allowed) 32 Mixed Exercises: Surds and Indices 33 34 Extension Material: 35 36 37 Solutions to all exercises: Exercise 1: Exercise 2: 38 Exercise 3: Exercise 4: Exercise 5: 39 Mixed Exercise: 40 Extension Material: 41

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