1. Dr J Frost (jfrost@tiffin.kingston.sch.uk) @DrFrostMaths
GCSE :: Laws of Indices
Dr J
Objectives: (a) Understand laws for multiplying power expressions, raising a power to a power and dealing with 0 or negative exponents. GCSE: (b) Deal with fractional exponents. (c) Deal with problems involving a mixture of bases.
Last modified: 6th January 2019

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3. 3 4 =3×3×3×3 ! Terminology “exponent” or “index” (plural “indices”) ?
“power”
3 4 =3×3×3×3
“Base”
The exponent tells us how many times the base appears in a product. ?
! We say this as “3 to the power of 4” or “3 raised to the power of 4” or “3 to the 4”.

4. More on notation
3 4
You may have heard the 4 referred to as the power. But ‘power’ refers to the whole expression 3 4 ; the 4 is the exponent!
Phrase
Precise meaning ?
“Powers of 2.”
Powers where the base is 2:
2 1 , 2 2 , 2 3 , 2 4 ?
“3 raised to the power of 2.”
3 2 . ‘Raised’ here means ‘turned into a power’. While “power of 2” might suggest the 2 is the ‘power’, it’s really short for “power with an exponent of 2”.

5. Understanding powers
When the exponent is a positive integer (whole number), it indicates how many times the base is repeated in the multiplication.
5 appears 2 times.
5 2 =𝟓×𝟓=𝟐𝟓 2 3 =𝟐×𝟐×𝟐=𝟖 3 4 =𝟑×𝟑×𝟑×𝟑=𝟖𝟏 2 5 =𝟐×𝟐×𝟐×𝟐×𝟐=𝟑𝟐 =𝟏𝟑 ? ? ? ? ?
It’s possible for the exponent to be fractional, 0 or negative. We’ll deal with these later!
Warning: Sometimes people incorrectly describe “ 4 3 ” as “4 multiplied by itself 3 times”. This would suggest there are 3 multiplications, but 4×4×4 actually only has 2 multiplications!

6. 𝑥 3 × 𝑥 2 =𝑥×𝑥×𝑥 × 𝑥×𝑥 = 𝑥 5 Multiplying powers 𝒙 𝒂 × 𝒙 𝒃 = 𝒙 𝒂+𝒃
How would we simplify this?
𝑥 3 means 3 𝑥’s multiplied together.
𝑥 3 × 𝑥 2
=𝑥×𝑥×𝑥 × 𝑥×𝑥
= 𝑥 5
In total we had 5 𝑥’s multiplied together.
! 1st law of indices:
𝒙 𝒂 × 𝒙 𝒃 = 𝒙 𝒂+𝒃
i.e. when we multiply two powers, we add the exponents.

7. Quickfire Questions
Your teacher will target various people. Do in your head! ?
𝑥 5 × 𝑥 4 = 𝒙 𝟗 𝑦 10 × 𝑦 10 = 𝒚 𝟏𝟎𝟎 𝑥 2 × 𝑥 3 × 𝑥 4 = 𝒙 𝟗 𝑝× 𝑝 4 = 𝒑 𝟓 𝑥× 𝑥 2 × 𝑥 9 = 𝒙 𝟏𝟐 𝑦 𝑘 × 𝑦 2 = 𝒚 𝒌+𝟐 𝑝 𝑎 × 𝑝 𝑛 = 𝒑 𝒂+𝒏 𝑞× 𝑞 𝑎 = 𝒒 𝟏+𝒂 ?
Fro Tip: When there is no exponent, you can raise the term to the power of 1:
𝑥→ 𝑥 1 ? ? ? ? ? ?

8. 𝑥 5 𝑥 3 = 𝑥×𝑥×𝑥×𝑥×𝑥 𝑥×𝑥×𝑥 = 𝑥 2 Dividing Powers 𝒙 𝒂 ÷ 𝒙 𝒃 = 𝒙 𝒂−𝒃
How would we simplify this?
𝑥 5 𝑥 3
= 𝑥×𝑥×𝑥×𝑥×𝑥 𝑥×𝑥×𝑥
Remember that we can simplify fractions by dividing the numerator and denominator by the same number (or term).
= 𝑥 2
! 2nd law of indices:
𝒙 𝒂 ÷ 𝒙 𝒃 = 𝒙 𝒂−𝒃
i.e. when we divide two powers, we subtract the exponents.

9. Quickfire Questions
Your teacher will target various people. Do in your head! ?
2 100 ÷ 2 2 = 𝟐 𝟗𝟖 𝑥 10 𝑥 3 = 𝒙 𝟕 𝑦 20 𝑦 −1 = 𝒚 𝟐𝟏 𝑥 15 ÷𝑥= 𝒙 𝟏𝟒 𝑥 3 × 𝑥 3 𝑥 2 = 𝒙 𝟔 𝒙 𝟐 = 𝒙 𝟒 ? ? ? ?

10. 𝑥 3 4 = 𝑥 3 × 𝑥 3 × 𝑥 3 × 𝑥 3 = 𝑥 12 Raising a Power to a Power
How would we simplify this?
𝑥 3 4
= 𝑥 3 × 𝑥 3 × 𝑥 3 × 𝑥 3
= 𝑥 12
! 3rd law of indices:
𝒙 𝒂 𝒃 = 𝒙 𝒂𝒃
i.e. when we raise a power to a power, we multiply the exponents.

11. Quickfire Questions
Your teacher will target various people. Do in your head! ?
𝑦 = 𝒚 𝟏𝟓 𝑥 3 × 𝑥 5 = 𝒙 𝟖 𝑝 = 𝒑 𝟓𝟔 𝑞 6 × 𝑞 9 = 𝒒 𝟏𝟓 𝑚 = 𝒎 𝟒𝟎 𝑢 8 × 𝑢 10 = 𝒖 𝟏𝟖 𝑎 −𝑏 −𝑐 = 𝒂 𝒃𝒄 ? ? ? ? ? ?

12. Mastermind Occupation: Student Favourite Teacher: Dr Frost
Specialist Subject: Laws of Indices
Insert your teacher’s name here

13. 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
Start Question > 26 23 ?
= 23

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

15. a b c
105
102
Start Question >
Start Question >
Start Question > ? ?
77 × 7-2 = 75 ?
= 103
(53)-2 = 5-6 e d f 87
8-2
Start Question >
Start Question >
Start Question > ? ?
_1_ 8 ?
= 89
8-2 × 84 = 82
2-3 = h g
Start Question >
Start Question > x9 ? =
x12
p2 x p= p3 ?
x-3

16. a b c
Start Question >
Start Question >
Start Question >
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
INSTANT DEATH g
Start Question > ?
(5-3)2 =5-6

17. a c b
Start Question >
Start Question > 50
5-2
Start Question >
51 x 52 x 53
= 56
(24 × 26)2 = 220 ? ?
= 52 ? e f
Start Question >
Start Question >
((41)2)3 = 46 ?
(23 × 23)3 = 218 ?

18. 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

19. What is the square root of 3 8 ?
Group Challenges
What is half of 2 7 ?
What is a ninth of ? 1 2
𝟐 𝟕 𝟐 = 𝟐 𝟕 𝟐 𝟏 = 𝟐 𝟔 ? ?
𝟑 𝟗𝟗 𝟗 = 𝟑 𝟗𝟗 𝟑 𝟐 = 𝟑 𝟗𝟕
What is a quarter of 4 𝑥 ?
What is the square root of 3 8 ? 3 4 ?
𝟒 𝒙 𝟒 = 𝟒 𝒙 𝟒 𝟏 = 𝟒 𝒙−𝟏
𝟑 𝟒 𝒃𝒆𝒄𝒂𝒖𝒔𝒆 𝟑 𝟒 𝟐 = 𝟑 𝟖 ?
4 𝑥 + 4 𝑥 + 4 𝑥 + 4 𝑥 = What is 𝑥? N ?
The LHS is 4∙ 4 𝑥 = 𝑥 = 4 𝑥+1 .
So 𝑥+1=16 → 𝑥=15

20. Exercise 1a
Questions on provided worksheet. 1
Simplify the following. 2
Simplify the following.
If 2 𝑥 × 2 𝑦 = 2 7 , what is 𝑥 in terms of 𝑦? 𝒙=𝟏𝟎−𝒚 3
𝑥 2 × 𝑥 3 = 𝒙 𝟓
𝑥 = 𝒙 𝟔
𝑚 20 𝑚 4 = 𝒎 𝟏𝟔
𝑦 8 × 𝑦 −2 = 𝒚 𝟔
𝑞× 𝑞 3 = 𝒒 𝟒
𝑝 = 𝒑 𝟏𝟎
𝑥× 𝑥 3 × 𝑥 5 = 𝒙 𝟗
𝑦 6 𝑦 2 = 𝒚 𝟒
𝑥 2 × 𝑥 𝑎 = 𝒙 𝟐+𝒂
𝑥 2 𝑦 = 𝒙 𝟐𝒚
𝑥 12 𝑥 3 = 𝒙 𝟗
𝑥 12 𝑥 = 𝒙 𝟏𝟏
𝑝 −𝑞 −𝑟 = 𝒑 𝒒𝒓
𝑤 4 𝑤 −4 = 𝒘 𝟖
𝑎 𝑏 𝑎 −𝑏 = 𝒂 𝟐𝒃 a ?
𝑦× 𝑦 10 𝑦 5 = 𝒚 𝟔
𝑥 2 × 𝑥 = 𝒙 𝟏𝟎
𝑝 12 𝑝 ×𝑝= 𝒑 𝟑𝟕
𝑥 10 × 𝑥 𝑥 = 𝒙 𝟒
𝑞 × 𝑞 5 𝑞 2 = 𝒒 𝟏𝟒
𝑥 2𝑦 𝑥 𝑦 3 = 𝒙 𝟑𝒚
𝑥× 𝑥× 𝑥 = 𝒙 𝟏𝟓 ? ? a b ? ? c ? b
Simplify the following:
2 𝑥 2 𝑦×3𝑥 𝑦 4 =𝟔 𝒙 𝟑 𝒚 𝟓 5 𝑝 3 𝑞 4 ×5𝑝𝑞=𝟐𝟓 𝒑 𝟒 𝒒 𝟓
10 𝑥 10 𝑦 2𝑥𝑦 =𝟓 𝒙 𝟗 𝑘 3 𝑚 𝑘 5 𝑚 = 𝟔 𝟓 𝒌 −𝟐 𝒎 𝟑
2𝑥𝑦×2 𝑥 𝑥 20 = 𝟏 𝟐 𝒙 −𝟗 𝒚 4 d ? ? ? c a ? e b ? ? f ? d c ? g ? ? ? e h d ? ? i f ? e ? ? j g ? k ? l ?
Exercise continues on next slide… m ? ? n o ?

21. Exercise 1a ? ? ? ? ? ? ? ? Questions on provided worksheet. N1 5 N2 6
[Edexcel GCSE June2003-6H Q17a]
If 𝑥= 2 𝑝 , 𝑦= 2 𝑞 , express the following in terms of 𝑥 and/or 𝑦:
(i) 2 𝑝+𝑞 = 𝟐 𝐩 × 𝟐 𝒒 =𝒙𝒚
(ii) 2 2𝑞 = 𝟐 𝒒 𝟐 = 𝒙 𝟐
(iii) 2 𝑝−1 = 𝟐 𝒑 𝟐 𝟏 = 𝒙 𝟐 N1 5
Solve 𝑥 = 𝑥
𝒙= 𝟒 𝟗
Given that 𝑥 = 𝑥 , express 4 𝑥 + 4 𝑥 as a single power of 4.
𝟒 𝒙 + 𝟒 𝒙 =𝟐× 𝟒 𝒙 = 𝟒 × 𝟒 𝒙 = 𝟒 𝟏 𝟐 × 𝟒 𝒙 = 𝟒 𝒙+ 𝟏 𝟐 ? ? ? ? N2 6
Simplify the following.
a) 3 𝑦 + 3 𝑦 + 3 𝑦 = 3 𝑦+1
b) 2 2𝑦 + 2 2𝑦 = 2 2𝑦+1
c) 2 𝑥 + 2 𝑥 2 = 2 2𝑥+2 ? ? ? ?

22. Zero and negative indices
Is there a pattern we can see that will help us out?
3 0
3 −1
33 = 27
32 = 9
31 = 3
30 = 1
3-1 =
3-2 = ÷3 ÷3
At this point, it doesn’t make sense to say “The product of -1 threes”. We’ll have to use a different approach! ? ÷3 1 3 ? 1 9 ?

23. Quickfire Questions
Your teacher will target various people. Do in your head!
6 −2 = 𝟏 𝟑𝟔 −1 =𝟏÷ 𝟐 𝟑 = 𝟑 𝟐 −1 = 𝟓 𝟒 −1 = 𝟖 𝟑 −1 =𝟐 −2 =𝟏÷ 𝟑 𝟓 𝟐 =𝟏÷ 𝟗 𝟐𝟓 = 𝟐𝟓 𝟗 −2 = 𝟑 𝟐 𝟐 = 𝟗 𝟒 −2 = 𝟒𝟗 𝟐𝟓 −3 = 𝟔𝟒 𝟐𝟕
4 −1 = 𝟏 𝟒 5 −1 = 𝟏 𝟓 =𝟏 7 −2 = 𝟏 𝟕 𝟐 = 𝟏 𝟒𝟗 8 −2 = 𝟏 𝟖 𝟐 = 𝟏 𝟔𝟒 2 −3 = 𝟏 𝟐 𝟑 = 𝟏 𝟖 =𝟏 4 −1 = 𝟏 𝟒 5 −3 = 𝟏 𝟓 𝟑 =𝟏𝟐𝟓 ? ?
A power of -1 therefore ‘flips’ (reciprocates) the fraction. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

24. Mini Exercise (Exercise 1b)
Questions on provided worksheet. 1
Determine the value of:
6 −1 = 𝟏 𝟔 =𝟏 8 −2 = 𝟏 𝟔𝟒 −2 = 𝟖𝟏 𝟔𝟒 N
[Edexcel GCSE June2003-6H Q17b]
Let 𝑥= 2 𝑝 , 𝑦= 2 𝑞 ,
If 𝑥𝑦 = 32 and 2𝑥 𝑦 2 =32, find the value of 𝑝 and the value of 𝑞 . ? a b ? ? c ? d ?
Dividing the second equation by first:
𝟐𝒚=𝟏 → 𝒚= 𝟏 𝟐 𝒙= 𝟑𝟐 𝒚 =𝟑𝟐÷ 𝟏 𝟐 =𝟔𝟒
Therefore 𝟐 𝒑 =𝟔𝟒 → 𝟔
𝟐 𝒒 = 𝟏 𝟐 → 𝒒=−𝟏 2 ?
𝒚=−𝟐 (as 𝟐 −𝟐 = 𝟏 𝟐 𝟐 = 𝟏 𝟒 )

25. 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 ?

26. And how could we prove this?
Fractional Indices
𝑥 = 𝑥
And how could we prove this? ?
𝒙 × 𝒙 =𝒙
But it’s also the case that:
𝒙 𝟏 𝟐 × 𝒙 𝟏 𝟐 = 𝒙 𝟏
by laws of indices.
So 𝒙 𝟏 𝟐 = 𝒙

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

28. Examples
= 𝟔𝟒 =𝟖 = 𝟑 𝟔𝟒 =𝟒 = 𝟖𝟏 =𝟗 = 𝟒 𝟖𝟏 =𝟑 ? =2 ? ?
3 𝑥 2 = 𝑥 = 𝑥 2 3 ? ? ?
− =−10 ?

29. Test Your Understanding So Far…
= 𝟑𝟔 =𝟔 = 𝟑 𝟖 =𝟐 − = 𝟓 −𝟑𝟐 =−𝟐 ? ? ?

30. What if the numerator is not 1?
‘Workings’-wise I usually skip his step.
= = 3 3 =27 ?
= 2 2 =4
…then just deal with what’s left.
Using denominator, do 5th power of 32 first to get 2 (but still have numerator left in the power to deal with) ?
Best to deal with negative in power first. Recall this does “1 over” the expression without the minus.
16 − 3 4 = = = 1 8

31. A few more examples ?
4 − 3 2 = 𝟏 𝟒 𝟑 𝟐 = 𝟏 𝟐 𝟑 = 𝟏 𝟖
− 2 3 = 𝟖 𝟐𝟕 𝟐 𝟑 = 𝟐 𝟑 𝟐 = 𝟒 𝟗
1 9 − 1 2 = 𝟗 𝟏 𝟐 =𝟑
Recall that “reciprocating” a fraction will cause it to flip. ? ?

32. Test Your Understanding?
= 𝟑 𝟐 =𝟗 9 − 3 2 = 𝟏 𝟗 𝟑 𝟐 = 𝟏 𝟑 𝟑 = 𝟏 𝟐𝟕 − 1 3 = 𝟔𝟒 𝟏𝟐𝟓 𝟏 𝟑 = 𝟒 𝟓 − 3 2 = 𝟗 𝟒 𝟑 𝟐 = 𝟑 𝟐 𝟑 = 𝟐𝟕 𝟖 ? ? ?

33. Exercise 2
Questions on provided worksheet.
− 1 3 = 3 4
64 − 1 3 = 1 4 ? 11
=10 ? 1 7 ? =5
− 3 2 = 64 27 2 ?
64 − 2 3 = 1 16 12 ? ? 8
16 −0.5 = 1 4 ? 3
− 3 4 = 𝟐𝟕 𝟖 ? =4 13 9 ?
27 − 2 3 = 1 9 4 ?
− 5 3 = 𝟐𝟒𝟑 𝟑𝟐
32 − 3 5 = 1 8 ? 14 ? 10
=16 5 ?
Write the following expression without using indices: 15
8 − 1 3 = 1 2 ? 6
𝑥 −0.5 = 1 𝑥 ?

34. 𝑎𝑏 2 = 𝑎 2 𝑏 2 𝑎+𝑏 2 = 𝑎+𝑏 𝑎+𝑏 = 𝑎 2 +2𝑎𝑏+ 𝑏 2
Applying indices to products ?
𝑎𝑏 2 = 𝑎 2 𝑏 2 ?
𝑎+𝑏 2 = 𝑎+𝑏 𝑎+𝑏 = 𝑎 2 +2𝑎𝑏+ 𝑏 2
The moral of the story:
Applying a power to a product applies the power to each term.
Applying a power to a sum does NOT apply power to each term. i.e. 𝑎+𝑏 𝑛 ≠ 𝑎 𝑛 + 𝑏 𝑛 in general.

35. Examples ?
2𝑥 2 =𝟒 𝒙 𝟐 ?
3 𝑥 2 𝑦 3 =𝟐𝟕 𝒙 𝟔 𝒚 𝟑
9 𝑥 =𝟑 𝒙 𝟑 ?
8 𝑥 6 𝑦 =𝟐 𝒙 𝟐 𝒚 𝟏 𝟑 ?

36. 9 𝑥 4 𝑦 6 3 𝑥 2 𝑦 1 2 Test Your Understanding Simplify 3 𝑥 2 𝑦 3 2?
9 𝑥 4 𝑦 6
Simplify 9 𝑥 4 𝑦 1 2 ?
3 𝑥 2 𝑦 1 2

37. Exercise 3 ? ? ? ? ? ? ? ? ? ? ? Questions on provided worksheet.
Simplify the following:
𝑥𝑦 2 = 𝒙 𝟐 𝒚 𝟐 3𝑥 2 =𝟗 𝒙 𝟐 𝑥 𝑦 = 𝒙 𝟐 𝒚 𝟒
2𝑐 𝑑 =𝟖 𝒄 𝟑 𝒅 𝟏𝟐
𝑎 𝑏 = 𝒂 𝟑 𝒃 𝟔
9 𝑎 =𝟑𝒂
16 𝑎 4 𝑏 =𝟒 𝒂 𝟐 𝒃 𝟑 𝟐
27 𝑎 9 𝑏 =𝟑 𝒂 𝟑 𝒃 𝟒 𝟑
8 𝑎 6 𝑏 =𝟒 𝒂 𝟒 𝒃 𝟖
16 𝑎 6 𝑏 =𝟔𝟒 𝒂 𝟗 𝒃 𝟏𝟖 27 𝑥 6 𝑦 5 =𝟗 𝒙 𝟒 𝒚 𝟓 𝟑 1 ? 2 ? ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 9 ? 10 ? 11

38. Law of Indices Backwards
This part of the topic is a bit more Further Mathsey…
Solve 𝑥 =27
The ‘thinking backwards’ method
The ‘cancelling the power’ method.
If I had some number to the power of 3 4 , what would I do to it?
Find the 4th root then cube it.
So going backwards from 27:
Cube root: 3
Raise to the power of 4: 81
What power should I raise both sides of the equation to ‘cancel’ the power?
𝒙 𝟑 𝟒 𝟒 𝟑 = 𝟐𝟕 𝟒 𝟑 𝒙= 𝟐𝟕 𝟒 𝟑 𝒙=𝟖𝟏 ? ?

39. Further Examples Solve 𝑥 − 2 3 =2 7 9 Solve 𝑦 −3 =3 3 8? ?
𝒙 − 𝟐 𝟑 = 𝟐𝟓 𝟗 𝒙= 𝟐𝟓 𝟗 − 𝟑 𝟐 = 𝟐𝟕 𝟏𝟐𝟓
𝒚 −𝟑 = 𝟐𝟕 𝟖 𝒚= 𝟐𝟕 𝟖 − 𝟏 𝟑 = 𝟐 𝟑

40. Test Your Understanding
Solve 𝑥 =9 ?
𝑥= 𝟗 𝟑 𝟐 =𝟐𝟕
Solve 𝑥 − 3 2 = 8 27 ?
𝑥= 𝟖 𝟐𝟕 − 𝟐 𝟑 = 𝟗 𝟒

41. Exercise 4 ? ? ? ? ? ? ? ? ? ? Questions on provided worksheet.
If 𝑥 =9, find 𝑥 𝒙=𝟐𝟕
Solve 𝑦 = 𝒚=𝟖
[AQA FM June 2012 Paper 1] 𝑥 =8 and 𝑦 −2 =
Work out the value of 𝑥 𝑦 𝒙÷𝒚=𝟒÷ 𝟐 𝟓 =𝟏𝟎
[AQA FM June 2013 Paper 1] Solve 𝑥 − 2 3 = writing your answer as a proper fraction. 𝟐𝟕 𝟓𝟏𝟐
[June 2013 Paper 2] 𝑝 −2 = 𝑞 6 × 𝑟 4
Write 𝑝 in terms of 𝑞 and 𝑟.
Give your answer in its simplest form.
𝑝= 𝒒 𝟔 × 𝒓 𝟒 − 𝟏 𝟐 = 𝒒 −𝟑 × 𝒓 −𝟐
[AQA FM Set 3 Paper 1] 𝑥 =6 and 𝑦 −3 =64
Work out the value of 𝑥 𝑦
𝒙÷𝒚=𝟑𝟔÷ 𝟏 𝟒 =𝟏𝟒𝟒
[AQA FM Set 1 Paper 2] You are given that 𝑥= 5 𝑚 and 𝑦= 5 𝑛 .
Write 5 𝑚+2 in terms of 𝑥. 𝟐𝟓𝒙
Write 5 𝑚−𝑛 in terms of 𝑥 and 𝑦 𝒙 𝒚
Write 5 3𝑛 in terms of 𝑦 𝒙 𝟑
Write 5 𝑚+𝑛 2 in terms of 𝑥 and 𝑦 𝒙𝒚 𝒐𝒓 𝒙 𝒚 6 ? 1 2 ? 3 ? 7 ? 4 ? ? ? ? 5 ? ?

42. 2, 8, 4 Changing bases ? What do you notice about all of the numbers:
They’re all powers of 2! We could replace the numbers with 2 1 , and so that we have a consistent base. ?

43. Changing bases Solve 2 𝑥 = 8 3 2 Solve 4 𝑥 = 2 10 2 2 𝑥 = 2 10
First convert everything to powers of 2.
𝟐 𝒙 = 𝟐 𝟑 𝟑 𝟐 𝟏 𝟐 𝟐 𝒙 = 𝟐 𝟗 ÷ 𝟐 𝟏 𝟐 = 𝟐 𝟖.𝟓 𝒙=𝟖.𝟓 ? ?
2 2 𝑥 = 2 10
2 2𝑥 = 2 10
𝑥=5
First convert everything to powers of 2… ? ? 3
If 2 2 = 2 𝑘 , determine 𝑘.
𝟐 𝟐 = 𝟐 𝟏 × 𝟐 𝟏 𝟐 = 𝟐 𝟑 𝟐 𝒌= 𝟑 𝟐 ?

44. Test Your Understanding1
If 9 3 = 3 𝑘 , find 𝑘.
𝟗 𝟑 = 𝟑 𝟐 × 𝟑 𝟏 𝟐 = 𝟑 𝟓 𝟐 𝒌= 𝟓 𝟐
Solve 3 𝑥 = 9 2𝑥−1
𝟑 𝒙 = 𝟑 𝟐 𝟐𝒙−𝟏 𝟑 𝒙 = 𝟑 𝟒𝒙−𝟐 𝒙=𝟒𝒙−𝟐 𝟐=𝟑𝒙 𝒙= 𝟐 𝟑 ? 2 ?

45. Exercise 5 1
Write as a single power of 2:
4 10 = 𝟐 𝟐 𝟏𝟎 = 𝟐 𝟏𝟎
8 𝑥 × 2 4 = 𝟐 𝟑𝒙+𝟒
= 𝟐 𝟒 𝟒 𝟑 = 𝟐 𝟏𝟔 𝟑
2 𝑥 × 4 𝑦 × 8 𝑧 ×16= 𝟐 𝒙+𝟐𝒚+𝟑𝒛+𝟒
2 2 = 𝟐 𝟏 × 𝟐 𝟏 𝟐 = 𝟐 𝟑 𝟐
= 𝟐 𝟐 𝟐 𝟏 𝟑 = 𝟐 𝟓 𝟑
× = 𝟐 𝟐 𝟑 × 𝟐 𝟑 𝟓 = 𝟐 𝟏𝟗 𝟏𝟓
[Edexcel GCSE(9-1) Nov F Q21c, Nov H Q6c Edited] 100 𝑎 × 1000 𝑏  can be written in the form  10 𝑤
Express 𝑤 in terms of 𝑎 and 𝑏.
𝒘=𝟐𝒂+𝟑𝒃
Solve for 𝑥:
8 𝑥 = 𝒙=𝟑
5 𝑥 = 𝒙= 𝟑 𝟐
4 𝑥 = 𝒙=𝟕
27 𝑥 = 𝒙=𝟏𝟎
4 2𝑥+1 = 8 2𝑥−1 𝒙= 𝟓 𝟐 4
[Edexcel GCSE(9-1) June H Q18]
× 2 𝑥 =
Work out the exact value of 𝑥.
𝒙=𝟏.𝟒𝟓
[Edexcel IGCSE Jan2017-4H Q16d]
= 3 𝑛
Work out the exact value of 𝑛.
𝒏=− 𝟒 𝟑
Solve
3 9 × 4 27 = 𝑥 3
𝟑 𝟐 𝟏 𝟑 × 𝟑 𝟑 𝟏 𝟒 = 𝟑 𝟏 𝒙 𝟑 𝟐 𝟑 × 𝟑 𝟑 𝟒 = 𝟑 𝟏 𝒙 𝟑 𝟏𝟕 𝟏𝟐 = 𝟑 𝟏 𝒙 𝟏𝟕 𝟏𝟐 = 𝟏 𝒙 𝒙= 𝟏𝟐 𝟏𝟕 ? a b ? ? ? c d ? 5 ? e ? f g ? ? 2 N ? ? 3 a ? b ? c ? d ? e ?