1. INDEX LAWS
Proof and tutorial

3. Index Laws
Learning Objectives:
• Able to estimate the answer to a question involving
indices
• Able to multiply terms with the same base
• Able to divide terms with the same base

4. Index Laws
• How can I simplify:45 x 43?
4 x 4 x 4 x 4 x x 4 x 4 x 4 = 48
4 9 6
• How can I simplify: 7 x 7 ?
7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x = 715
x7 7

5. Law 1 : Multiplication
26 x 24 = 210
24 x 22 = 26
35 x 37 = 312
General Rule:
When multiplying terms with the same base, you
add the powers
EXAMPLE: 47 x 45 = 47+5 =
am x an = am+n 412

6. Index Laws
• How can I simplify 95 ÷ 93?
9x9x9x9x9
= 9 x 9 = 92
9x9x9
• How can I simplify 117 ÷ 112?
11 x 11 x 11 x 11 x 11 x 11 x
11 = 115
11 x 11

7. Law 2 : Division
26 24 = 22
25 ÷ 22 = 23
35 ÷ 37 = 3-2
General Rule
When dividing terms with the same base, you
subtract the powers
EXAMPLE: 47 ÷ 45 = 47-5 = 42
am an = am-n

8. Index Laws
• How can I simplify (4 3) 4?
4 x 4 x x 4 x 4 x x 4 x 4 x x 4 x 4 x = 412
5
4 4
• How can I simplify (72) ? 4 4
7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 710

9. Law 3 : Brackets
(26)2 = 26 x 26 = 212
(35)3 = 35 x 35 x 35 = 315
General Rule:
When raising a power to another power, you
multiply the powers
EXAMPLE: (47) 5= 47 x 5 = 435
(am)n = am x n

10. Index Laws
• Simplify the following:
73 x 78
75

11. Level 8 17/01/2013
Index Laws
Learning Objectives:
• Able to multiply terms with the same base
• Able to divide terms with the same base
• Able to simplify terms with negative
indices

12. Index Laws
• How can I simplify 5a3 x 6a7?
5xaxaxa 6xaxaxaxaxaxa
x
xa
5x6xaxaxaxaxaxaxaxax=
axa 30a10

13. Algebra and Indices
• 9d6 x 8d3
• 12e10 x 6e4
• 24f8 ÷ 6f4
• 60g12 ÷ 5g3

14. Starter
•

15. Index Laws
Learning Objectives:
• Able to multiply and divide terms with the same base
• Able to raise a power to another power
• Able to simplify terms with negative
indices

16. Algebra and Indices
2
• (7c3) = 7 x c x c x cx 7 x c x c x c
=7x7xcxcxcxcxcxc
= 49c6
3
• (4j8)
4
• (5k-3)

17. Law 4: Negative Indices
25 = 32
24 = 16 General Rule
23 = 8
22 = 4 1
21 = 2 a-n = n
a
20 = 1
2-1 = ½
2-2 = ¼
2-3 = ⅛
-4 = 1
2 16

18. Law 5: Power 0
25 = 32
24 = 16
23 = 8 GENERAL
22 = 4 RULE:
21 = 2
20 = 1 Any term to the
2-1 = ½ power 0 is
2-2 = ¼ equal to 1.
2-3 = ⅛
-4 = 1
2 16

19. Negative Indices
• 6-2
• 7-3
• 9m-4

20. Simplify the following, leaving your answers
in index notation.
1. (38) 4 = 332 2. 7f3 x 9f9 = 63f12
3. 124 x = 1214 4. (6m5 )3 =
1210 216m15
5. 25 ÷ 2-3 = 28 6. 42g7 ÷ = 6g5
7g2
7. 39 x 36 = 310 8. 116 x 116 = 117
35 112 x 113
Solve the following.
9. 7-2 10. 3-3

21. Negative Indices
•

22. More difficult examples
• Simplify: 43 x 47 6p8 x 3p3
46 9p4 x p7
• Solve:
83 -1)3
(4
85