1. Dr J Frost (jfrost@tiffin.kingston.sch.uk)
GCSE: Indices
Dr J Frost
Last modified: 23rd August 2013

2. Starter 𝒙 𝟑 × 𝒙 𝟒 𝒙 −𝟐 𝒙 𝟕 𝒙 𝟑 𝒙 𝟏 𝟐 𝒙 𝟏 𝒙 𝟎 𝒙 𝟑 𝟒 𝒙 𝟑 × 𝒙 −𝟒 𝟏 𝒙 𝟕
In pairs or otherwise, try and match the blue and orange cards.
𝒙 𝟑 × 𝒙 𝟒
𝒙 −𝟐
𝒙 𝟕 𝒙 𝟑
𝒙 𝟏 𝟐
𝒙 𝟏
𝒙 𝟎
𝒙 𝟑 𝟒
𝒙 𝟑 × 𝒙 −𝟒 𝟏
𝒙 𝟕
𝒙 𝟒
𝟏 𝒙 𝒙
𝟏 𝒙 𝟐 𝒙
𝒙 𝟏𝟐

3. A reminder of the Laws of Indices? ?
𝑎 𝑏 × 𝑎 𝑐 = 𝑎 𝑏+𝑐
𝑎 0 =1
𝑎 𝑏 𝑎 𝑐 = 𝑎 𝑏−𝑐 ? ?
𝑎 1 =𝑎 ?
𝑎 𝑏 𝑐 = 𝑎 𝑏𝑐
𝑎 −𝑏 = 1 𝑎 𝑏 ?

4. Examples ?
2 𝑥 𝑦 = 2 𝑥𝑦
= 9 7
4 4 × = = 4 −1 = 1 4
2 𝑦 −1 = 2 𝑦
2𝑦 −1 = 1 2𝑦 ? ? ? ?

5. Mastermind Occupation: Student Favourite Teacher: Dr Frost
Specialist Subject: Laws of Indices

6. Instructions: Everyone starts by standing up
Instructions: Everyone starts by standing up. You’ll get a question with a time limit to answer. If you run out of time or get the question wrong, you sit down. The winner is the last man standing.
Warmup:
Start Question > ?
Start Question > ?
23 × 24 = 27
(23)4 = 212 26 23
Start Question > ?
= 23

7. a b c
911 92
Start Question >
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47 × 43 = 410 ? ?
= 99
(35)2 = 310 e d f 57 53
Start Question >
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= 54 ?
74 × 76 = 710
(46)3 = 418 ? h g
Start Question >
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_1_ 2 ? ?
(22)2 = 24
2-1 =

8. a b c
105
102
Start Question >
Start Question >
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77 × 7-2 = 75 ?
= 103
(53)-2 = 5-6 e d f 87
8-2
Start Question >
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_1_ 8 ?
= 89
8-2 × 84 = 82
2-3 = h g
Start Question >
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_1_ 16 ? ?
50 = 1
4-2 =

9. a b c
Start Question >
Start Question >
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9-2 ? ?
4-2 × 4-2 = 4-4 ?
= 90 = 1
(3-2)-2 = 34 e d f
Start Question >
101
10-3
Start Question >
Start Question > ?
= 104 ?
14 × 16 = 110 = 1 h g
Start Question >
Start Question >
_1_ 56
_1_ 27 ? ?
(5-3)2 =5-6 =
3-3 =

10. a b c
Start Question > 50
5-2
Start Question >
Start Question > ? ?
50 = 1 ?
= 52
(30)2 = 1 e d f
Start Question >
51 x 52 x 53
= 56
Start Question >
Start Question >
(24 × 26)2 = 220 ?
((41)2)3 = 46 ? ? h g
Start Question >
Start Question >
_1_ 81 ?
(23 × 23)3 = 218 ?
3-4 =

11. a b c
Start Question >
Start Question >
Start Question >
47 × 43 42
(35)4 33
(73)3
(72)3 ? ? ?
= 48
= 317
= 73 d e f
Start Question >
Start Question >
Start Question >
58 × 58
51 × 5-1
((32)2)2 32
(71)3
(72)1 ? ? ?
= 516
= 36
×74 = 75

12. Exercises Simplify the following. 2 𝑥 2 = 2 𝑥−1 ? ? 𝑎 3 × 𝑎 5 = 𝑎 8 ?
5 𝑥 𝑥 ×5= 5 𝑥 2 +1 13
2 𝑥 2 = 2 𝑥−1 ? ?
𝑎 3 × 𝑎 5 = 𝑎 8 1 ? 8
𝑧 8 𝑧 = 𝑧 24 14 ?
𝑎 = 𝑎 15 ? 2
𝑥 𝑥 2 = 𝑥 6 9 ?
𝑚 2 −5 = 𝑚 −10 or 1 𝑚 10 ?
5 3𝑥 5 4𝑥 2 = 𝑥 3 15 ?
𝑦 9 × 𝑦 − 𝑦 = 𝑦 2
𝑥 9 𝑥 3 = 𝑥 6 10 ? ? 4
2 𝑦 −7 = 2 𝑦 7 ?
3 2𝑥 = 3 2𝑥−2 11 16 5 ?
4 𝑥 + 4 𝑥 + 4 𝑥 + 4 𝑥 =4⋅ 4 𝑥 = 4 𝑥+1 ?
𝑦 8 𝑦 −2 = 𝑦 10 6 ? 17
𝑥 10 𝑥 2 = 𝑥 18
5 3𝑥 5 𝑥 2 = 5 4𝑥 ? 12 ? ? 7
7 𝑥 × 7 3 = 7 𝑥+3

13. And how could we prove this?
Fractional Indices
𝑥 = 𝑥
And how could we prove this?

14. Fractional Indices
𝑥 = 3 𝑥 ?
𝑥 1 𝑛 = 𝑛 𝑥 ?

15. Examples =8
𝑥 3 = 𝑥 = 𝑥 3 2 ? ? =4 =2 ? ? =9
= = ? ? =3
− =−10 ? ?

16. 9 3 2 =27 16 − 3 4 = 1 8 32 2 5 =4 9 − 1 2 = 1 3 Harder Examples ? ? ?
=27
16 − 3 4 = 1 8 ? ? =4 ?
9 − 1 2 = 1 3 ?

17. Exercises
=10 ?
−64 − 1 3 =− 1 4 ? 1 7 =5 ? 2
−64 − 2 3 = 1 16 8 ?
16 −0.5 = 1 4 ? 3 =4 9 ?
27 − 2 3 = 1 9 ? 4
32 − 3 5 = 1 8 ? 10
=16 5 ? 11
Write the following expression without using indices:
8 − 1 3 = 1 2
𝑥 −0.5 = 1 𝑥 ? ? 6

18. 𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑎 𝑏 −𝑛 = 𝑏 𝑛 𝑎 𝑛
Applying indices to products and fractions ?
𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛
𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 ?
𝑎 𝑏 −𝑛 = 𝑏 𝑛 𝑎 𝑛 ?

19. Applying indices to products and fractions?
2𝑥 2 =4 𝑥 2
1 2 −5 =32 ?
9 𝑥 =3 𝑥 3 ?
= 3 2 ?
3 𝑥 2 𝑦 3 =9 𝑥 6 𝑦 3 ?
= 1 32
− 2 3 = 4 9 ? ?

20. 27 8 − 2 3 𝑓𝑙𝑖𝑝 8 27 2 3 𝑟𝑜𝑜𝑡 2 3 2 𝑝𝑜𝑤𝑒𝑟 4 9 ‘Flip Root Power’ method
− 𝑓𝑙𝑖𝑝 𝑟𝑜𝑜𝑡 𝑝𝑜𝑤𝑒𝑟 ? ? ?

21. Exercises
Evaluate:
= 3 2
Simplify:
= 8 27 ? ? 7 8 1
𝑎 𝑏 = 𝑎 3 𝑏 6 ?
9 𝑎 =3𝑎 2 ?
2 3 −3 = 27 8 ? 9
16 𝑎 4 𝑏 =4 𝑎 2 𝑏 3 2 ? 3
27 𝑎 9 𝑏 =3 𝑎 3 𝑏 4 3 ?
5 6 −1 = 6 5 ? 4 10
8 𝑎 6 𝑏 =4 𝑎 4 𝑏 8 5 ?
− 1 3 = 3 4 11 ?
16 𝑎 6 𝑏 =64 𝑎 9 𝑏 18 6 ?
− 3 2 = 64 27 12 ?

22. Skill 3: Changing bases
What do you notice about all of the numbers:
2, 8, 4
They’re all powers of 2! We could replace the numbers with 2 1 , and so that we have a consistent base. ?

23. Skill 3: Changing bases Solve 4 𝑥 = 2 10 Solve 2 𝑥 = 8 3 2?
2 2 𝑥 = 2 10
2 2𝑥 = 2 10
𝑥=10 ?
𝑥= 17 2 ? ?

24. Difficult GCSE question
𝑥= 2 𝑝 , 𝑦= 2 𝑞
a) Express in terms of 𝑥 and/or 𝑦.
i) 2 𝑝+𝑞 = 2 𝑝 2 𝑞 =𝑥𝑦
ii) 2 2𝑞 = 2 𝑞 2 = 𝑦 2
iii) 2 𝑝−1 = 2 𝑝 = 𝑥 2
b) Given that:
𝑥𝑦=32 2𝑥 𝑦 2 =32
find the value of 𝑝 and 𝑞.
𝑝=6, 𝑞=−1 ? ? ? ? ?

25. Exercises
Solve for 𝑥:
𝑥=3 ? 1
8 𝑥 = 2 9
2 𝑥 =2 2
𝑥= 3 2 ? 2
4 𝑥 = 3
𝑥=7 ? 4
2 𝑥 + 2 𝑥 = 2 19
𝑥=18 ?
27 𝑥 =
𝑥=10 ? 5 ?
4 2𝑥+1 = 8 2𝑥−1
𝑥= 5 2 6