1. 4 Laws of Integral Indices
4.1 Simplifying Algebraic Expressions Involving Indices
4.2 Zero and Negative Integral Indices
4.3 Simple Exponential Equations
4.4 Different Numeral Systems
4.5 Inter-conversion between Different Numeral Systems

2. 4.1 Simplifying Algebraic Expressions Involving Indices

3. 4.1 Simplifying Algebraic Expressions Involving Indices
A. Law of Index of (am)n

4. Example 1T 4 Laws of Integral Indices Solution:
Simplify each of the following expressions.
(a) (q3) (b) (q2  q5)2
Solution:
(a) (q3)8  q3  8
(b)

5. Example 2T 4 Laws of Integral Indices Solution:
Simplify 64y  8x  42y, where x and y are positive integers.
Solution:
64y  8x  42y

6. 4.1 Simplifying Algebraic Expressions Involving Indices
B. Law of Index of (ab)n

7. Example 3T 4 Laws of Integral Indices Solution:
Simplify each of the following expressions.
(a) (11u2) (b) (3b4)3
Solution:
(a) (11u2)2 
(b) (3b4)3 

8. Example 4T 4 Laws of Integral Indices Solution:
Simplify each of the following expressions.
(a) [p2 (q3)2] (b) (–5h5x3)2
Solution:
(a)
(b)

9. 4.1 Simplifying Algebraic Expressions Involving Indices
a n
b
C. Law of Index of
is undefined.

10. Example 5T 4 Laws of Integral Indices Solution:
Simplify each of the following expressions (where n  0 and d  0).
(a) (b)
Solution:

11. Example 6T 4 Laws of Integral Indices Solution:
Simplify each of the following expressions (where h  0, k  0, and v  0).
(a) (b)
Solution:
(a)
(b)

12. Example 7T 4 Laws of Integral Indices Solution:
Expand each of the following expressions.
(a) (b) (c)
Solution:
(a)
(b)

13. Example 7T 4 Laws of Integral Indices Solution:
Expand each of the following expressions.
(a) (b) (c)
Solution:
(c)

14. 4.1 Simplifying Algebraic Expressions Involving Indices

15. 4.1 Simplifying Algebraic Expressions Involving Indices

16. 4.2 Zero and Negative Integral Indices
00 is undefined. Thus, a0 = 1 for a  0.

17. 4.2 Zero and Negative Integral Indices
It is also true for

18. 4.2 Zero and Negative Integral Indices

19. Example 8T 4 Laws of Integral Indices Solution:
Find the values of the following expressions without using a calculator.
(a) 30  25 (b) (7)3  (2)0 (c) 53  (10)2
Solution:

20. Example 9T 4 Laws of Integral Indices Solution:
Simplify the following expressions (where a  0, b  0 and c  0) and express the answers with positive indices.
(a) (a)4 (b) (c)
Solution:
(a)
(b)
(c)

21. Example 10T 4 Laws of Integral Indices Solution:
Simplify the following expressions (where u  0 and s  0) and express the answers with positive indices.
(a) (u2)2(u1)5 (b) (3s1)  (s)4
Solution:

22. Example 11T 4 Laws of Integral Indices Solution:
Simplify the following expressions (where all variables are non-zero) and express the answers with positive indices.
(a) (b)
Solution:
(a)

23. Example 11T 4 Laws of Integral Indices Solution:
Simplify the following expressions (where all variables are non-zero) and express the answers with positive indices.
(a) (b)
Solution:
(b)

24. Example 12T 4 Laws of Integral Indices Solution:
Let x be a positive integer. Simplify the following expressions.
(a) (b)
Solution:
(a)
(b)

25. 4.3 Simple Exponential Equations

26. Example 13T 4 Laws of Integral Indices Solution:
Solve the following exponential equations.
(a) 103k  (b) 2k  1 (c) 6k 
Solution:
(a) 103k  1000
(b) 2k  1
(c) 6k 
103k  103
2k  20
6k 
3k  3
k  0 
k  1 
6k  63
k  3 

27. Example 14T 4 Laws of Integral Indices Solution:
Solve the following exponential equations.
(a) (b)
Solution:
(b)  

28. 4.4 Different Numeral Systems
In daily life, we often use the word ‘weight’ which refers to the meaning of mass.

29. 4.4 Different Numeral Systems
A. Denary System

30. 4.4 Different Numeral Systems A. Denary System
The idea of place values is also applicable to digits beyond the decimal points. For example:
0.25(10) = 2  10–1 + 5  10–2

31. Example 15T 4 Laws of Integral Indices Solution:
Express the following denary numbers in expanded form with base 10.
(a) (10) (b) (10)
Solution:
(a)
 80  9  0.6  0.04
(b)
 2  0.004

32. 4.4 Different Numeral Systems
B. Binary System

33. 4.4 Different Numeral Systems
B. Binary System

34. Example 16T 4 Laws of Integral Indices Solution:
(a) Express 1  22  0   20 as a binary number.
(b) Express 1  23  1  21  1  20 as a binary number.
Solution:
(a)
(b)

35. 4.4 Different Numeral Systems
C. Hexadecimal System

36. 4.4 Different Numeral Systems
C. Hexadecimal System

37. Example 17T 4 Laws of Integral Indices Solution:
(a) Express 2  162  14  161  1  160 as a hexadecimal number.
(b) Express  162  10 as a hexadecimal number.
Solution:
(a)
(b)

38. 4.5 Inter-conversion between Different Numeral Systems
A. Converting Binary/Hexadecimal Numbers into Denary Numbers

39. Example 18T 4 Laws of Integral Indices Solution:
Convert the following binary numbers into denary numbers.
(a) 111(2)
(b) 1001(2)
Solution:

40. Example 19T 4 Laws of Integral Indices Solution:
Convert the following hexadecimal numbers into denary numbers.
(a) 66(16)
(b) 12C(16)
Solution:

41. 4.5 Inter-conversion between Different Numeral Systems
B. Converting Denary Numbers into Binary/Hexadecimal Numbers

42. Example 20T 4 Laws of Integral Indices Solution:
Convert the denary number 33(10) into a binary number.
Solution: 2 33 2 16 … 1 2 8 … 4 … 2 2 … 2 1 …
33(10)  (2)

43. Example 21T 4 Laws of Integral Indices Solution:
Convert the denary number 530(10) into a hexadecimal number.
Solution: 16
530 33 … 2 16 2 … 1
530(10)  212(16)