1. Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter Summary Mathematics in Workplaces 2.4Different Numeral Systems 2.5Inter-conversion between Different Numeral Systems

2. P. 2 Biologist In the 1840’s, biologists found that all plants and animals, including humans, are made up of cells. Cells are created from cell division. Each time a cell division takes place, a parent cell divides into 2 daughter cells. Solving exponential equations like 2 n  2 15 can help biologists determine the growth rate of cells. Mathematics in Workplaces

3. P. 3 2.1Simplifying Algebraic Expressions Involving Indices A. Law of Index of (a m ) n Suppose m and n are positive integers, we have  a mn For any positive integers h and k, we have a h  a k  a h  k. If m and n are positive integers, then (a m ) n  a mn.

4. P. 4 Simplify each of the following expressions. (a)(q 3 ) x (b)(q 3 ) 8 (c)(q 2y ) 5 2.1Simplifying Algebraic Expressions Involving Indices A. Law of Index of (a m ) n Example 2.1T Solution: (a)(q 3 ) x  q 3  x  q3x  q3x (b)(q 3 ) 8  q 3  8  q 24 (c)(q 2y ) 5  q 2y  5  q 10y

5. P. 5 2.1Simplifying Algebraic Expressions Involving Indices B. Law of Index of (ab) n Suppose n is a positive integer, we have  anbn anbn Group the terms of a and b separately. If n is a positive integer, then (ab) n  a n b n.

6. P. 6 2.1Simplifying Algebraic Expressions Involving Indices B. Law of Index of (ab) n Example 2.2T Simplify each of the following expressions. (a)(11u 2 ) 2 (b)(3b 4 ) 3 Solution: (a)(11u 2 ) 2  11 2 u 2  2  121u 4 (b)(3b 4 ) 3  3 3 b 4  3  27b 12

7. P. 7 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of is undefined. If n is a positive integer, then, where b  0. When a fraction is multiplied by itself n times, where b  0 and n is any positive integer, we can simplify the expression as follows:

8. P. 8 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Example 2.3T Simplify each of the following expressions. (a), n  0 (b), d  0 Solution:

9. P. 9 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Example 2.4T Simplify each of the following expressions. (a), h and k  0 (b), v  0 For any positive odd integer m, (  1) m   1. For any positive even integer n, (  1) n  1. Solution:

10. P. 10 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Example 2.5T Simplify 64 y  8 x  4 2y. Solution: 64 y  8 x  4 2y Change the numbers to the same base before applying the laws of indices, i.e., write 64  2 6, 8  2 3 and 4  2 2.

11. P. 11 2.2Zero and Negative Integral Indices A. Zero Index In Book 1A, we learnt that a m  a n  a m  n for m  n. Consider the case when m  n: a m  n  a 0 However, if we calculate the actual value of the expression 3 2  3 2, 3 2  3 2  9  9  1 Hence, we define the zero index of any non-zero number as follows: We can conclude that 3 0  1. For example, 3 2  3 2  3 2  2  3 0. If a  0, then a 0  1. 0 0 is undefined.

12. P. 12 B. Negative Integral Indices Consider a m  a n  a m  n. If m  n, then m  n is negative. The expression a m  n has a negative index. For example, 5 2  5 3  5 2  3  5  1. However, 5 2  5 3  25  125 . We can conclude that 5  1 . Hence, we define the negative index of any non-zero number as follows: 2.2Zero and Negative Integral Indices If a  0 and n is a positive integer, then. 0  n is undefined.

13. P. 13 Example 2.6T B. Negative Integral Indices 2.2Zero and Negative Integral Indices Find the values of the following expressions without using a calculator. (a)3 0  2 5 (b)(  7)  3  (  2) 0 (c)5  3  (  10)  2 Solution:

14. P. 14 2.2Zero and Negative Integral Indices Summarizing the previous results, we have the following laws of integral indices. If m and n are integers, then 1.a m  a n  a m  n 2.a m  a n  a m  n (where a  0) 3.(a m ) n  a mn 4.(ab) n  a n b n 5.(where b  0) 6.a 0  1(where a  0) 7.(where a  0) B. Negative Integral Indices

15. P. 15 Example 2.7T 2.2Zero and Negative Integral Indices B. Negative Integral Indices Simplify the following expressions and express the answers with positive indices. (a)(u 2 ) 2 (u  1 ) 5, u  0(b)(3s  1 )  (  s)  4, s  0 Solution: Since it is stated that each answer should be written with positive indices, it is incorrect to express the answer as u  1.

16. P. 16 Example 2.8T 2.2Zero and Negative Integral Indices B. Negative Integral Indices Simplify the following expressions and express the answers with positive indices. (a), y  0(b), p, q and r  0 Solution:Alternative Solution:

17. P. 17 Example 2.8T 2.2Zero and Negative Integral Indices B. Negative Integral Indices Simplify the following expressions and express the answers with positive indices. (a), y  0(b), p, q and r  0 Solution: Since it is stated that each answer should be written with positive indices, it is incorrect to express the answer in terms of q  10.

18. P. 18 The variable x of this equation appears as an index. Such equations are called exponential equations. Method of solving exponential equations: First express all numbers in index notation with the same base. For example,2 x  8 2 x  2 3 x  3 2.3Simple Exponential Equations Consider the equation 2 x  8. Then simplify the expression using laws of integral indices if necessary. For example, (9 t ) 2  81 9 2t  9 2 2t  2 t  1 (a m ) n  a mn Express both sides as powers of 2 Equate the indices on both sides

19. P. 19 2.3Simple Exponential Equations Example 2.9T Simplify the following exponential equations. (a)10 3k  1000(b)2 k  1(c)6 k  Solution: (a)10 3k  1000 10 3k  10 3 3k  3 k  1 (b)2 k  1 2 k  2 0 k  0 k   3 (c)6 k  6k6k 6 k  6  3

20. P. 20 2.3Simple Exponential Equations Example 2.10T Simplify the following exponential equations. (a) (b)2 x  1  5  2 x  28 Solution: (b) 2 x  1  5  2 x  28 2  2 x  5  2 x  28 (2  5)  2 x  28 7  2 x  28 2 x  4 2 x  2 2 x  2 Express all numbers in index notation with the same base. Apply the techniques of solving linear equations with one unknown.

21. P. 21 The most commonly used numeral system today is the denary system. Numbers in this system are called denary numbers. The denary system consists of 10 basic numerals: ‘0’, ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’, ‘7’, ‘8’ and ‘9’. Consider the expanded form of 236 with base 10: 236 = 2  10 2 + 3  10 1 + 6  10 0 The numbers 10 2, 10 1 and 10 0 are the place values of the corresponding positions/digits of a number. The place values of numbers in this system differ by powers of 10. A. Denary System 2.4Different Numeral Systems

22. P. 22 Another commonly used numeral system is the binary system. Numbers in this system are called binary numbers. The binary system consists of only 2 numerals: ‘0’ and ‘1’. For example, the expanded form of 1011 (2) is: 1011 (2) = 1  2 3 + 0  2 2 + 1  2 1 + 1  2 0 The numbers 2 3, 2 2, 2 1 and 2 0 are the place values of the corresponding positions/digits of a number. The place values of the digits in a binary number differ by powers of 2. 2.4Different Numeral Systems B. Binary System

23. P. 23 Another commonly used numeral system is the hexadecimal system. Numbers in this system are called hexadecimal numbers. The hexadecimal system consists of 16 numerals and letters: ‘0’, ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’, ‘7’, ‘8’, ‘9’, ‘A’, ‘B’, ‘C’, ‘D’, ‘E’ and ‘F’. The letters A to F represent the values 10 (10) to 15 (10) respectively. For example, the expanded form of 13A (16) is: 13A (16) = 1  16 2 + 3  16 1 + 10  16 0 The numbers 16 2, 16 1 and 16 0 are the place values of the corresponding positions/digits of a number. The place values of the digits in a hexadecimal number differ by powers of 16. 2.4Different Numeral Systems C. Hexadecimal System

24. P. 24 2.4Different Numeral Systems C. Hexadecimal System Example 2.11T (a)Express 1  2 2  0  2 1  1  2 0 as a binary number. (b)Express 4  10 2  9  10 1  0  10 0 as a denary number. Solution:

25. P. 25 It can be done by summing up all the terms in the expanded form. A. Convert Binary/Hexadecimal Numbers into Denary Numbers 2.5Inter-conversion between Different Numeral Systems We can make use of the expanded form to convert binary/hexadecimal numbers into denary numbers.

26. P. 26 2.5Inter-conversion between Different Numeral Systems A. Convert Binary/Hexadecimal Numbers into Denary Numbers Example 2.12T Convert the following binary numbers into denary numbers. (a)111 (2) (b)1001 (2) Solution:

27. P. 27 2.5Inter-conversion between Different Numeral Systems A. Convert Binary/Hexadecimal Numbers into Denary Numbers Example 2.13T Convert the following hexadecimal numbers into denary numbers. (a)66 (16) (b)12C (16) Solution:

28. P. 28 2.5Inter-conversion between Different Numeral Systems B. Convert Denary Numbers into Binary/Hexadecimal Numbers It can be done by considering all the remainders in the short division. We make use of division to convert denary numbers into binary/hexadecimal numbers.

29. P. 29 2.5Inter-conversion between Different Numeral Systems B. Convert Denary Numbers into Binary/Hexadecimal Numbers Example 2.14T Convert the denary number 33 (10) into a binary number. Solution: 233 16…1 8…0 2 2 24…0 22…0 1…0 33 (10)  100001 (2)

30. P. 30 2.5Inter-conversion between Different Numeral Systems B. Convert Denary Numbers into Binary/Hexadecimal Numbers Example 2.15T Convert the denary number 530 (10) into a hexadecimal number. Solution: 16530 33…216 2…1 530 (10)  212 (16)

31. P. 31 Chapter Summary 2.1 Simplifying Algebraic Expressions Involving Indices For positive integers m and n, 1.(a m ) n  a mn. 2.(ab) n  a n b n. 3., where b  0.

32. P. 32 Chapter Summary 2.2 Zero and Negative Integral Indices For any non-zero number a and positive integer n, 1.a 0  1. 2.a  n .

33. P. 33 2.3 Simple Exponential Equations When solving exponential equations, first express all numbers in index notation with the same base, then simplify using the laws of integral indices. Chapter Summary

34. P. 34 SystemBinaryDenaryHexadecimal Digits used 0, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Place values 2 0, 2 1, …10 0, 10 1, …16 0, 16 1, … 2.4 Different Numeral Systems Chapter Summary

35. P. 35 Inter-conversion of numbers can be done by division or expressing them in the expanded form. 2.5 Inter-conversion between Different Numeral Systems Chapter Summary

36. A. Law of Index of (a m ) n 2.1Simplifying Algebraic Expressions Involving Indices Follow-up 2.1 Simplify each of the following expressions. (a)(y 6 ) 2 (b)(y d ) 2 (c)(y 3 ) 2m Solution: (a)(y 6 ) 2  y 6  2  y 12 (b)(y d ) 2  y d  2  y2d  y2d (c)(y 3 ) 2m  y 3  2m  y6m  y6m

37. B. Law of Index of (ab) n 2.1Simplifying Algebraic Expressions Involving Indices Simplify each of the following expressions. (a)(4m) 3 (b)(13a 7 ) 2 Solution: (a)(4m) 3  4 3 m 3  64m 3 (b)(13a 7 ) 2  13 2 b 7  2  169a 14 Follow-up 2.2

38. 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Follow-up 2.3 Solution: Simplify each of the following expressions. (a)(b), q  0

39. 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Follow-up 2.4 Solution: Simplify each of the following expressions. (a), a and b  0 (b), x  0

40. 2.1Simplifying Algebraic Expressions Involving Indices C. Law of Index of Follow-up 2.5 Simplify each of the following expressions. (a)25 3x  125 y  5 4y (b)16 2x  8 4x  2 3y Solution:

41. Find the values of the following expressions without using a calculator. (a)10 0  9 2 (b)(  4)  1  5 0 (c)4  3  6  1 Follow-up 2.6 B. Negative Integral Indices 2.2Zero and Negative Integral Indices Solution:

42. Follow-up 2.7 B. Negative Integral Indices 2.2Zero and Negative Integral Indices Simplify the following expressions and express the answers with positive indices. (a)(h 4 )  1 (h  2 ) 3, h  0(b)(  k)  5  (k 4 ), k  0 Solution:

43. Follow-up 2.8 B. Negative Integral Indices 2.2Zero and Negative Integral Indices Solution:Alternative Solution: Simplify the following expressions and express the answers with positive indices. (a), h  0(b), b, c and d  0

44. Follow-up 2.8 B. Negative Integral Indices 2.2Zero and Negative Integral Indices Simplify the following expressions and express the answers with positive indices. (a), h  0(b), b, c and d  0 Solution:Alternative Solution:

45. Follow-up 2.9 2.3Simple Exponential Equations Simplify the following exponential equations. (a)2 3x  64(b)8 2y  1(c)3 y  Solution: (a) 2 3x  64 2 3x  2 6 3x  6 x  2 (b)8 2y  1 8 2y  8 0 y  0 y 4 y 4 (c)3 y  3y3y 3 y  3  4 2y  0

46. Follow-up 2.10 2.3Simple Exponential Equations Simplify the following exponential equations. (a) 16 y  1  2 6  2y (b)5 x  1  2  5 x  75 Solution: (a) 16 y  1  2 6  2y (2 4 ) y  1  2 6  2y 2 4y  4  2 6  2y 4y  4  6  2y 2y  2 y  1 (b) 5 x  1  2  5 x  75 5  5 x  2  5 x  75 (5  2)  5 x  75 3  5 x  75 5 x  25 5 x  5 2 x  2

47. Follow-up 2.11 2.4Different Numeral Systems C. Hexadecimal System (a)Express 8  10 2  5  10 1  3  10 0 as a denary number. (b)Express 14  16 2  0  16 1  1  16 0 as a hexadecimal number. Solution:

48. Follow-up 2.12 2.5Inter-conversion between Different Numeral Systems A. Convert Binary/Hexadecimal Numbers into Denary Numbers Convert the following binary numbers into denary numbers. (a)101 (2) (b)10011 (2) Solution:

49. 2.5Inter-conversion between Different Numeral Systems A. Convert Binary/Hexadecimal Numbers into Denary Numbers Follow-up 2.13 Convert the following hexadecimal numbers into denary numbers. (a)70 (16) (b)5F3 (16) Solution:

50. 2.5Inter-conversion between Different Numeral Systems B. Convert Denary Numbers into Binary/Hexadecimal Numbers Follow-up 2.14 226 13…0 6…1 23…0 26 (10)  11010 (2) Convert the following denary numbers into binary numbers. (a) 26 (10) (b)35 (10) 2 2 1…1 Solution: (a) 235 (b) 17…1 8…12 4…02 2…02 1…0 35 (10)  100011 (2) 2

51. 2.5Inter-conversion between Different Numeral Systems B. Convert Denary Numbers into Binary/Hexadecimal Numbers Follow-up 2.15 Convert the following denary numbers into hexadecimal numbers. (a) 83 (10) (b)418 (10) Solution: 1683 5…3 83 (10)  53 (16) (a) 2418 (b) 26…2 1…10 418 (10)  1A2 (16) 2