The document discusses index laws and meanings in algebra. It covers: - Adding and subtracting like terms, and inability to add unlike terms - Index laws for multiplication and division of terms with exponents - Meanings of exponents as they relate to fractions, roots, and powers - Examples of expanding and simplifying expressions using index laws - Solutions to exercises involving application of index laws

1. Methods In Algebra
Like terms can be added or subtracted, unlike
terms cannot.

2. Index Laws
a m  a n  a m n

3. Index Laws
a m  a n  a m n
a m  a n  a mn

4. Index Laws
a m  a n  a m n
a m  a n  a mn
a 
m n
 a mn

5. Index Laws
a m  a n  a m n
a m  a n  a mn
a 
m n
 a mn
a0  1

6. Index Meaning
 : top of the fraction

7. Index Meaning
 : top of the fraction
 : bottom of the fraction

8. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
b
x

9. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
b
x root

10. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
x b
root  b xa
OR
  x
b a

11. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
x b
root  b xa
OR
  x
b a
e.g. (i ) x 3 

12. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
x b
root  b xa
OR
  x
b a
3
1
e.g. (i ) x  3
x

13. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
x b
root  b xa
OR
  x
b a
3
1
e.g. (i ) x  3 (ii ) a 5b 7 
x

14. Index Meaning
 : top of the fraction
 : bottom of the fraction
a
power
x b
root  b xa
OR
  x
b a
3
1 a5
e.g. (i ) x  3 (ii ) a 5b 7  7
x b

15. 3
(iii ) x  4 a 9b  2 
4

16. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b

17. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4

18. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x

19. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3

20. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
3
(vi ) x 
2

21. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
3
(vi ) x 
2
x3

22. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
3
(vi ) x 
2
x3
 x2 x

23. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
3
(vi ) x 
2
x3
 x2 x
x x

24. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
see
3 3
(vi ) x 
2
x 3 OR x 
2
 x2 x
x x

25. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
see think
3 3 1
(vi) x 
1
OR x 
2 3
x 2
x 2
 x2 x
x x

26. 3 3a 9
(iii ) x  4 a 9b  2  4 2
4 4x b
1
(iv) x 4 4
x
2
(v ) y 3 3
x2
see think
3 3 1
(vi) x 
1
OR x 
2 3
x 2
x 2
 x2 x x x
x x 1
1
x and x 2

27. 27
(vii ) m 4


28. 27
 m m
4 64 3
(vii ) m

29. 27
 m m
4 64 3
(vii ) m
1 7
1 6 500  28 6 69
(viii ) n p q c r 
2

30. 27
 m m
4 64 3
(vii ) m
1 7
1 6 500  28 6 69
(viii ) n p q c r 
2 2

31. 27
 m m
4 64 3
(vii ) m
1 7
1 6 500  28 6 69
(viii ) n p q c r 
2 2 n6

32. 27
 m m
4 64 3
(vii ) m
1
1 6 500  28 6 69
7
p 500
(viii ) n p q c r 
2 2 n6

33. 27
 m m
4 64 3
(vii ) m
1
1 6 500  28 6 69
7
p 500
(viii ) n p q c r 
2 2 n 6 28 q

34. 27
 m m
4 64 3
(vii ) m
1
1 6 500  28 6 69
7
p 500 c 6 c
(viii ) n p q c r 
2 2 n 6 28 q

35. 27
 m m
4 64 3
(vii ) m
1
1 6 500  28 6 69
7
p 500 c 6 c r 69
(viii ) n p q c r 
2 2 n 6 28 q

36. 27
 m m
4 64 3
(vii ) m
1 6 500  28 6 69
1 7
p 500 c 6 c r 69
(viii ) n p q c r 
2 2 n 6 28 q
2
2
(ix)   
3

37. 27
 m m
4 64 3
(vii ) m
1 6 500  28 6 69
1 7
p 500 c 6 c r 69
(viii ) n p q c r 
2 2 n 6 28 q
2 2
2
(ix)    3
  
3  2

38. 27
 m m
4 64 3
(vii ) m
1 6 500  28 6 69
1 7
p 500 c 6 c r 69
(viii ) n p q c r 
2 2 n 6 28 q
2 2
2
(ix)    3
  
3  2
9

4

39. 27
 m m
4 64 3
(vii ) m
1 6 500  28 6 69
1 7
p 500 c 6 c r 69
(viii ) n p q c r 
2 2 n 6 28 q
2 2
2
(ix)    3
  
3  2
9

4
Exercise 1A; 1c, 2d, 3b, 4d, 5b, 6ad, 7bc, 8a, 9b, 10d, 11cf,
12ac, 13bd, 15, 17, 18*
Exercise 6A; 1adgi, 2behj, 3ace, 4ace, 5bdfh, 6ace, 7adgj,
8behj, 9bd