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If a is a real number and n is a positive integer, then a base n = a.a.a.a … a (n times) index The number a is called the base and n is called the index, and a n is read as ‘ a to the power of n ‘ 1 RULES OF INDICES 1. a a a m n 2. a a a m 3. ( a n m ) a n m n mn mn Notes 1. 2. 3. a a 0 n 1 n 1 1 n a a a m n a m a b n n 4. a 5. ab 6. an a n b b n n n n 3 Example 1 Evaluate each of the following without using calculator. (a) 8 2 3 8 (b) 27 - 2 3 4 Example 2 Simplify each of the following. 3 2 (a) (y ) y (c) x y 2 2 x y 2 3 (b) ( t ) ( t ) 3 5 2 5 Example 3 Simplify 4 6 3 form 2 m n 3 8 1 2 in the 12 and 3 6 SURDS OBJECTIVES (a)Explain the meaning of a surd and its conjugate, and to carry out algebraic operations on surds (b) State the rules of indices Surds An irrational number and expressed in terms of root sign Positive integer n a Real number Note: a is not a perfect square, a > 0 Let’s pronounce correctly a is nth root of a a is square root of a 3 a is cube root of a 4 a is fourth root of a n 9 Rules of surds 1 n 1. n a a 2. n a a 3. n ab a b n a na n b b 4. 5. n n m n n a mn a 10 Algebraic Operations on Surds a) Multiplication a× a =a a× b = ab Example 1: When unlike surds are multiplied together, the product is a surd. a) 3 × 3 3 3 =3 b) 5 × 2 5 2 = 10 b) Division a =1 a÷ a = a a a÷ b = Example 2: a) 2÷ 2 b) 21 ÷ 5 = b 2 2 21 = 5 = a b =1 21 = 5 c) Addition and Subtraction a c ±b c = a ± b c Example 3: i) 4 3 +2 3 = 4 + 2 3 =6 3 ii) 4 3 -2 3 = 4 -2 3 =2 3 d) Expansion of Surds a± b = 2 a± b a± b = a a± a b± b a± b b = a ± b ± 2 ab Expansion of Surds (Alternatively) : a± b = a 2 2 ±2 a b + = a ±2 ab +b b 2 a± b Example 4: i) 2+ 3 2 2 = = a 2 2 2 ±2 a b + +2 2 3 + = 2 + 3 + 2 2 3 =5 +2 6 b 3 2 2 Rationalising the Denominator Surd Conjugate 2 + 3 -4 - 5 The conjugate of the surd a+ b is a- b Example 5 : Rationalise the denominators of each of the following fractions. 8 8 2 8 2 (a) =4 2 = × = 2 2 2 2 Solving Equations Involving Surds Example 6: Solve the following equations. Give your answer in the set form. (a) 6x +1 -5 = 0 (b) 2 x 4 - x -1 4