ADDITIONAL MATHEMATICS FORM 4 CHAPTER 4: INDICES, SURDS, AND LOGARITHMS AHMAD SARHAN BIN MOHD SALMIZI 2023277904
Learning Outcome 1.Express the laws of indices 2.Able to solve the algebraic problems using laws of indices 3.Explain the meaning of a surd and its conjugate 4.Express the laws of surds 5.Perform algebraic operations on surd’s questions 6.Express the laws of logarithms 7.Able to change the base of logarithm 8.Solve the logarithm’s questions using laws of logarithms 9.Solve the problems involving the applications of indices, surds, and logarithms
4.1 LAWS OF INDICES Positive / negative / zero / fractional indices • 4 = × × × • − = 1 • 0 = 1 • = (√ ) Laws of indices • × = + • ÷ = − • ( ) = • × = () • = ( ) Problem-solving involving indices • If = , then = • If = , then = Example question: Malacca Trial SPM Question 2022 (Paper 1) 1. (a) Simplify 1 2 × 3 − 1 3 (2 marks) = 1 2 × 3 − 1 3 = 3 ( 1 2 +(− 1 3 ) ) = 3 1 6 (b) Show that 5 +1 − 5 −1 + 5 is divisible by 29 for all positive integers . (2 marks) = 5 +1 − 5 −1 + 5 = 5 (5) − 5 (5 −1 ) + 5 = 5 (5 − 1 5 + 1) = 5 ( 29 5 ) = 29(5 −1 ) ∴ 29(5 −1 ) can be divisible by 29 for all positive integers . Simplify the algebraic expressions involving indices Example Remarks 3 3−2 ÷ 3 −3 = 3 3−2−(−3) = 3 3−2−+3 = 3 2+1 Use law of indices • ÷ = −
4.2 LAWS OF SURDS Laws of Surds • √ × √ = • √ × √ = √ • √ √ = √ • ( + √)( − √) = 2 − When a number cannot be reduced by eliminating its root (Whether it is a square root or a cube root and so on) then it is a surd. Conjugate Surd Surd Conjugate Surd √ √ √ + √ √ − √ √ − √ √ + √ Example question: Johor (Tangkak) Trial SPM Question 2022 (Paper 2) 2. (a) The area of a rectangle is given by (3 + √2) 2 . If the length is (2 − √2) 2 , find its width in the form of ( + √2) 2 . (4 marks) = × (3 + √2) = × (2 − √2) = 3+√2 2−√2 = 3+√2 2−√2 × 2+√2 2+√2 = 6+3√2+2√2+2 4−2 = 8+5√2 2 = 4 + 5 2 √2
4.3 LAWS OF LOGARITHMS = write as = , while > 0 and ≠ 1 Laws of Logarithms • If = , then = • 10 can be written as or • can be written as • = + • = − • = • = • = 1 • = 1 • 1 = 0 Example question: Selangor Trial SPM Question 2022 Set A (Paper 1) 1. (b) Evaluate (√9)()(3). (3 marks) (√9)()(3) = ( 9 √ ) () ( 1 3 ) = ( 9 1 2 ) () ( 1 3 ) = ( 9 1 2 ) () ( 1 3 ) = (2 9 ) () ( 1 3 ) = (29) ( 1 3 ) = (23 2 ) ( 1 3 ) = ((2 × 2)3) ( 1 3 ) = 4
4.4 APPLICATIONS OF INDICES, SURDS, AND LOGARITHMS Solving problems involving indices, surds, and logarithms. Example problem: Epidemiologists found that the transmission of Covid-19 virus among humans spread over an () = 1000 × 5 0.2 acre, with being the number of weeks after the initial observation was made. a) Find the original area of Covid-19 virus infection. b) Find the area of Covid-19 virus infection after i. 5 weeks ii. 10 weeks. c) How long does it take for the Covid-19 virus to spread to an area of 50000 acres? Solution a) () = 1000 × 5 0.2 (0) = 1000 × 5 0.2(0) = 1000 × 5 0 = 1000 × 1 = 1000 acres b) i. () = 1000 × 5 0.2 (5) = 1000 × 5 0.2(5) = 1000 × 5 1 = 5000 acres ii. () = 1000 × 5 0.2 (10) = 1000 × 5 0.2(10) = 1000 × 5 2 = 25000 acres c) () = 1000 × 5 0.2 50000 = 1000 × 5 0.2 50 = 5 0.2 50 = 5 0.2 50 = 0.2 5 50 5 = 0.2 0.2 = 2.4307 = 12.15 weeks
~THANK YOU~ SARHAN SALMIZI @ahan_0917 @sahe0917 I hope this flipbook can help students to further understand more about the concepts of indices, surds, and logarithms. Also, students should be able to express every law of indices, surds, and logarithms in order to be experts in this topic. Lastly, students should be able to solve any kind of problems involving indices, surds, and logarithms.