Download GCSE: Indices

GCSE: Indices
Skipton Girls’ High School
Objectives: Understand and use the laws of indices. Change the base, and
apply knowledge to more complex equations.
Terminology Recap
!
“Base”
?
“Power” or “index”
?
(plural: “indices”)
or
“exponent”
4
3
! We say this as “3 to the power of 4” or “3
raised to the power of 4” or “3 to the 4”.
The whole expression is sometimes (confusingly)
referred to as a ‘power’ or ‘power expression’.
Multiplying Power Expressions with the Same Base
5
3
4
3
×
How would I write this multiplication out in full?
Therefore, how could I write the result of this multiplication
in the form 3k?
9
3
?
1st Law of Indices
!
b
a
×
c
a
=
b+c
a
?
Dividing power expressions with the same base
7
4
2
4
How would I write this multiplication out in full?
Therefore, how could I write the result of this multiplication
in the form 4k?
5
4
?
2nd Law of Indices
!
b
a
c
a
=
?
b-c
a
Raising a power to a power
!
2
3
(4 )
How would I write this multiplication out in full?
Therefore, how could I write the result of this multiplication
in the form 4k?
6
4
?
3rd Law of Indices
!
b
c
(a )
=
bc
a
?
Zero and negative indices
Is there a pattern we can
see that will help us out?
0
3
-1
3
At this point, it doesn’t
make sense to say “Multiply
3 by itself negative 1 times”.
We’ll have to use a different
approach!
33 = 27
2
3 =9
1
3 =3
30 = 1?
3-1
=
1?
3
3-2 = 19?
Final Laws of Indices
!
1
a
=a
0
a =1
1
-b
a = ab
Challenges
1
What is half of
27 ?
𝟕
𝟕
2
𝟐
𝟐
= ?𝟏 = 𝟐𝟔
𝟐
𝟐
3
What is a
quarter of 4𝑥 ?
What is a ninth of
399 ?
𝟗𝟗
𝟗𝟗
𝟑
𝟑
= ?𝟐 = 𝟑𝟗𝟕
𝟗
𝟑
4
What is the square
8
root of 3 ?
𝟒𝒙 𝟒𝒙
= 𝟏? = 𝟒𝒙−𝟏
𝟒
𝟒
5
4𝑥 + 4𝑥 + 4𝑥 + 4𝑥 = 416
𝟑𝟒
𝒃𝒆𝒄𝒂𝒖𝒔𝒆? 𝟑𝟒
𝟐
What is x?
The LHS is 4 ∙ 4𝑥 = 41 4x = 4x+1 .
So 𝑥 + 1 = 16? → 𝑥 = 15
= 𝟑𝟖
Exercise 1
Please ensure you write out the question.
3
1 Simplify the following.
a) 𝑥 2 × 𝑥 3 = 𝑥 5?
b) 𝑥 2 3 = 𝑥 6?
c) 𝑦 8 × 𝑦 −2 = 𝑦?6
d) 𝑞 × 𝑞 3 = 𝑞 4?
e) 𝑥 2 × 𝑥 𝑎 = 𝑥 2+𝑎
?
2
𝑦
2𝑦
f) 𝑥
= 𝑥?
g)
𝑥 12
𝑥3
4
a) 3𝑦 + 3𝑦 + 3𝑦 = 3𝑦+1
?
2𝑦
2𝑦
2𝑦+1
b) 2 + 2 = 2 ?
c) 2𝑥 + 2𝑥 2 = 22𝑥+2
?
6
= 𝑦?
b) 𝑥 2 × 𝑥 3
𝑥 2𝑦
𝑥𝑦
3
2
10
= 𝑥?
c)
= 𝑥 3𝑦
?
d) 𝑥 × 𝑥 × 𝑥 3
2 2
Simplify the following.
= 𝑥 15
?
=
2
24 𝑥
𝟒
?𝟗
𝒙=
1
b) 4−3 = 64?
c) 1−1 = 1 ?
d) 80 = 1 ?
1
e) 7 × 7−3 = 49?
f) 3−2 −2 = 81?
g) 10−4 = 0.0001
?
2 Simplify the following.
𝑦5
2𝑥 5
23
1
𝑥 12
11
ℎ)
= 𝑥?
𝑥
4
𝑤
8
i) 𝑤 −4 = 𝑤?
a) 𝑦 ×
N1 Solve
a) 3−4 = 81?
= 𝑥 9?
𝑦 10
Evaluate the following (i.e. give as a
fraction or integer with no power)
N2 Given that 𝑥 12 =
𝑥, express 4𝑥 +
4𝑥 as a single
power of 4.
𝟒𝒙 + 𝟒𝒙 = 𝟐 × 𝟒𝒙
= 𝟒 × 𝟒𝒙
?
𝟏
= 𝟒𝟐 × 𝟒𝒙
𝟏
= 𝟒𝒙+𝟐
Changes of Base
Solve the following equation.
10
9
=3
𝑥
Express as a single power.
𝑥
4 ×8
𝑥+1
(Hint: can we express 9 as a power of 3
perhaps?)
2 10
3
=3
20 ? 𝑥
3 =3
𝑥 = 20
𝑥
𝟐 𝒙
𝟑 𝒙+𝟏
𝟐
× 𝟐
= 𝟐𝟐𝒙 ×? 𝟐𝟑𝒙+𝟑
= 𝟐𝟓𝒙+𝟑
The strategy therefore is to find what both bases are a power of (e.g. 4 and 8
are both powers of 2), and replace them as such.
A few more examples…
Solve
Solve
125𝑥+1 = 25𝑥
4𝑥+1 = 83
𝟐 𝒙+𝟏
𝟐
= 𝟐
𝟐𝟐𝒙+𝟐 = 𝟐𝟗
?𝟗
𝟐𝒙 + 𝟐 =
𝟕
𝒙=
𝟐
𝟑 𝟑
Express as a single power of 3:
37 × 93 = 37 × 32
= 37 × 36?
3
𝒙+𝟏
𝟑
𝟓
𝒙
𝟐
𝟓
=
𝟓𝟑𝒙+𝟑 = 𝟓𝟐𝒙
𝟑𝒙 + 𝟑 = ?𝟐𝒙
𝒙 = −𝟑
Test Your Understanding
8
2
x
4
=
4
27
x = 4?
x
8
=
12
4
x = 8?
Express as a single power:
?
𝟑𝒙 × 𝟐𝟕𝒙−𝟏 = 𝟑𝟒𝒙−𝟑
=
x
9
x = 6?
x
3
=
100
27
?
x = 300
Express as a single power:
?
𝟒𝟏𝟎 × 𝟖𝒙 = 𝟐𝟐𝟎+𝟑𝒙
Exercise 2
Please ensure you write out the question.
Solve the following.
1
2
3
4
5
6
Express as a single power.
2𝑥 = 410
3 𝑥 = 95
5𝑥+1 = 253
43 = 8 𝑥
9𝑥 = 274
𝒙 = ?𝟐𝟎
𝒙 = ?𝟏𝟎
𝒙 =?𝟓
𝒙 =?𝟐
𝒙 =?𝟔
𝟒𝟎
𝒙 =?
𝟑
83𝑥 = 1630
Solve the following.
7 9 𝑥+1 = 27 𝑥−1
8
9
810 = 42𝑥+1
125
𝑥+1
1−𝑥
= 25
10
16𝑦 = 32𝑦−1
11
43𝑥−1 = 81−3𝑥
𝒙 =?𝟓
𝒙 =?𝟕
𝟏
𝒙 =?−
𝟓
𝒚 =?𝟓
𝟏
𝒙 =?
𝟑
12
13
14
15
N
?
4𝑥 × 23 = 𝟐𝟐𝒙+𝟑
?
3𝑥 × 27𝑦 = 𝟑𝒙+𝟑𝒚
125𝑥
𝟑𝒙−𝟐𝒚
=
𝟓
?
25𝑦
16𝑥+1 × 32𝑥−1 = 𝟐𝟗𝒙−𝟏
?
3
𝑥+2
4
𝟏𝟓𝒙−𝟔
?
=
𝟐
2−𝑥
8
Reminder of the Laws of Indices
?
𝑎𝑏 × 𝑎𝑐 = 𝑎𝑏+𝑐
𝑎0 = 1 ?
𝑎𝑏
𝑏−𝑐
?
=
𝑎
𝑐
𝑎
𝑎1 = 𝑎 ?
𝑎
𝑏 𝑐
= 𝑎𝑏𝑐?
−𝑏
1
= 𝑏?
𝑎
𝑎
Matching Activity
𝟑
𝟒
𝒙 ×𝒙
𝟏
𝒙
−𝟐
𝟎
𝒙𝟕
𝒙𝟑
𝟒
𝟑
𝒙
𝒙
𝒙
𝟏
𝒙𝟕
𝒙𝟒
𝒙
𝟏
𝒙𝟐
𝒙
𝟏
𝒙𝟐
𝒙𝟑 × 𝒙−𝟒
𝟏
𝒙
𝒙𝟏𝟐
Examples
2𝑥 𝑦
?
𝒙𝒚
𝟐
=
?
2𝑥 × 2𝑦 = 𝟐𝒙+𝒚
98
= 𝟗𝟕?
9
44 × 4 𝟒𝟓
? 𝟒−𝟏
=
=
42 3
𝟒𝟔
𝟏
𝟏
−3
4 = 𝟑 =?
𝟒
𝟔𝟒
𝟏
𝟏
−5
2 = 𝟓 =?
𝟑𝟐
𝟐
1
3
2
3
−1
−2
𝟑
= =𝟑
𝟏
𝟑
=
𝟐
𝟐
Recall “reciprocating” a
fraction (i.e. 1 over) flips the
fraction. Can you think why?
?
𝟗
=
?
𝟒
Test Your Understanding
𝑥5
7
× 𝑥7
𝑥2
5−1
𝟏
= ?
𝟓
−3
𝟏
?
=
𝟐𝟕
3
2
5
5
2
= 𝒙𝟒𝟎?
−1
−3
𝟓
= ?
𝟐
𝟖
?
=
𝟏𝟐𝟓
Exercise 1
12
Simplify the following.
1
𝑎3 × 𝑎5 = 𝑎8?
2
𝑎3
3
𝑚2 −5
𝑥9
4
5
6
𝑥3
5
7
?
= 𝑎15
=
8
𝑚−10
?or
𝑥6
= ?
𝑦8
10
=
𝑦
?
𝑦 −2
?
7𝑥 × 73 = 7𝑥+3
1
𝑚10
9
10
11
2𝑥
?
= 2𝑥−1
2
13
𝑥2 4
6?
=
𝑥
𝑥2
𝑦 9 × 𝑦 −2
𝑦3 4
5𝑥 𝑥 × 5
2
= 5𝑥 ?+1
3
5
14
𝟓
= ?
𝟑
−2
6
𝟒𝟗?
=
7
𝟑𝟔
2
1
−2
5 =
25
1
−1
6 =
6
= 𝑦?2
16
N
15
𝑎
𝑏
−1
3
2
−3
−3
𝟖
= ?
𝟐𝟕
𝒃𝟑
= 𝟑?
𝒂
4𝑥 + 4𝑥 + 4𝑥 + 4𝑥
= 4 ⋅ 4𝑥 = 4𝑥+1
?
Fractional Indices
1
𝑥2
= 𝑥
And how could we prove this?
𝒙× 𝒙=𝒙
But it’s also the case that:
𝟏
𝒙𝟐
by laws of indices.
𝟏
𝟐
So 𝒙 = 𝒙
×
𝟏
𝒙𝟐
? = 𝒙𝟏
Fractional Indices
1
𝑥3
1
𝑥𝑛
=
3
=
𝑛
?
𝑥
𝑥
?
Examples
1
642
=8
?
1
643
=4
1
812
=9
1
814
= 3?
?
?
0.25
16
3
7
2
= 2?
1 2
?
3
7
=
=
−1000
1
3
?
= −10
2
73
Test Your Understanding So Far…
1
?
2
36 = 𝟑𝟔 = 𝟔
1
𝟑
?
83 = 𝟖 = 𝟐
1
𝟓
−32 5 = −𝟑𝟐? =
−𝟐
What if the numerator is not 1?
3
92
=
2
325
1 3
92
?
3
= 3 = 27
…then just deal with what’s left.
2
=2 =4
3
−
16 4
?
Using denominator, do 5th power of 32 first to
get 2 (but still have numerator left in the
power to deal with)
=
1
Best to deal with negative in power first. Recall this
does “1 over” the expression without the minus.
3
4
1
1
= ?3 =
2
8
A few more examples
3
−
4 2
27
8
1
9
=
𝟏
𝟑
𝟒𝟐
2
−
3
𝟏
𝟏
= ?𝟑 =
𝟐
𝟖
𝟐
𝟑
𝟖
=
𝟐𝟕
1
−
2
=
𝟏
𝟗𝟐
=𝟑
?
Recall that “reciprocating” a
fraction will cause it to flip.
𝟐
= ?
𝟑
𝟐
𝟒
=
𝟗
Test Your Understanding
2
273
3
−
9 2
𝟐 ?
=𝟑 =𝟗
𝟏
𝟏
𝟏
= 𝟑 = ?𝟑 =
𝟑
𝟐𝟕
𝟗𝟐
125
64
4
9
3
−
2
1
−
3
𝟔𝟒
=
𝟏𝟐𝟓
𝟗
=
𝟒
𝟑
𝟐
𝟏
𝟑
?
𝟑
= ?
𝟐
𝟒
=
𝟓
𝟑
𝟐𝟕
=
𝟖
Exercise 2
1
?
1000.5 = 10
2
1
1253
3
16−0.5
4
5
6
−
27
4
83
−
8
2
3
= 5?
1
= ?
4
1
= ?
9
7
−64
8
−64
9
2
325
10
3
−5
32
1
−3
2
−3
1
= −?
4
11
1
= ?
16
= 4?
1
= ?
8
12
13
14
64
27
9
16
16
81
8
27
1
−3
3
−2
3
−4
5
−3
3
=?
4
64
= ?
27
𝟐𝟕
= ?
𝟖
𝟖𝟏
= ?
𝟏𝟔
= 16?
1
3
1
= ?
2
15
Write the following expression without using
indices:
𝑥 −0.5
1
= ?
𝑥
Applying indices to products and fractions
𝑎𝑏
𝑎+𝑏
2
2
2? 2
=𝑎 𝑏
= 𝑎 + 𝑏 ?𝑎 + 𝑏
2
2
= 𝑎 + 2𝑎𝑏 + 𝑏
The moral of the story:
1. Applying a power to a product applies the power to each term.
2. Applying a power to a sum does NOT apply power to each term.
i.e. 𝑎 + 𝑏 𝑛 ≠ 𝑎𝑛 + 𝑏𝑛 in general.
Examples
2𝑥
2
3𝑥 2 𝑦
?𝟐
= 𝟒𝒙
3
1
6
9𝑥 2
6
8𝑥 𝑦
= 𝟐𝟕𝒙?𝟔 𝒚𝟑
=
1
3
𝟑
?
𝟑𝒙
=
𝟏
?𝟐 𝟑
𝟐𝒙 𝒚
Test Your Understanding
Simplify 3𝑥 2 𝑦 3
2
9𝑥 4 𝑦? 6
Simplify 9𝑥 4 𝑦
1
2
1
2 ?2
3𝑥 𝑦
Law of Indices Backwards
(IGCSE Further Maths)
3
4
Solve 𝑥 = 27
The ‘thinking backwards’ method
The ‘cancelling the power’ method.
(DO THIS!)
3
If I had some number to the power of 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
3
the equation to ‘cancel’ the 4 power?
𝟒
𝟑 𝟑
𝒙𝟒
=
𝟒
?
𝒙 = 𝟐𝟕𝟑
𝒙 = 𝟖𝟏
𝟒
𝟐𝟕𝟑
Further Examples
Solve 𝑥
2
−
3
=
7
2
9
𝟐𝟓
𝒙 =
𝟗
𝟑
−
?
𝟐𝟓 𝟐
𝟐𝟕
𝒙=
=
𝟗
𝟏𝟐𝟓
𝟐
−𝟑
Solve 𝑦
−3
=
3
3
8
𝟐𝟕
=
𝟖
? −𝟏
𝟐𝟕 𝟑 𝟐
𝒚=
=
𝟖
𝟑
𝒚−𝟑
Test Your Understanding
2
3
Solve 𝑥 = 9
𝑥=
Solve 𝑥
3
−2
=
𝟑
𝟗𝟐? =
𝟐𝟕
8
27
𝟖?
𝑥=
𝟐𝟕
𝟐
−𝟑
𝟗
=
𝟒
Exercises
2𝑐𝑑 4 3
1 [June 2012 Paper 1] Simplify
2
𝑎𝑏2
3
1
2 2
9𝑎
4
5
6
7
3
=
?
𝟖𝒄𝟑 𝒅𝟏𝟐
= 𝒂𝟑 𝒃?𝟔
= 𝟑𝒂 ?
𝑝=
1
𝟑
4 3 2
𝟐 𝟐
16𝑎 𝑏
= 𝟒𝒂 𝒃
1
𝟒
9 4 3
𝟑 𝟑
27𝑎 𝑏
= 𝟑𝒂 𝒃
2
6 12 3
8𝑎 𝑏
= 𝟒𝒂𝟒 𝒃𝟖
3
6
12
16𝑎 𝑏 2 = 𝟔𝟒𝒂𝟗 𝒃𝟏𝟖
?
?
?
?
2
3
8 If 𝑥 = 9, find 𝑥.
4
3
𝟏
−
𝟒
×𝒓 𝟐
? 𝒒−𝟑 × 𝒓−𝟐
=
−3
13 [Set 3 Paper 1] 𝑥 = 6 and 𝑦 = 64
𝑥
Work out the value of 𝑦
𝟏
𝒙 ÷ 𝒚 = 𝟑𝟔?÷ = 𝟏𝟒𝟒
𝟒
?
𝒚=𝟖 ?
9 Solve 𝑦 = 16
3
25
−2
10 [June 2012 Paper 1] 𝑥 2 = 8 and 𝑦 = 4 .
𝑥
𝒒𝟔
1
2
𝒙 = 𝟐𝟕
Work out the value of 𝑦
−2
6
4
12 [June 2013 Paper 2] 𝑝 = 𝑞 × 𝑟
Write 𝑝 in terms of 𝑞 and 𝑟.
Give your answer in its simplest
form.
𝟐
𝒙 ÷ 𝒚 = 𝟒?÷ 𝟓 = 𝟏𝟎
11 [June 2013 Paper 1] Solve 𝑥
2
−3
answer as a proper fraction.
=
𝟐𝟕
𝟓𝟏𝟐
1
79
?
writing your
14 [Set 1 Paper 2] You are given that
𝑥 = 5𝑚 and 𝑦 = 5𝑛 .
(a) Write 5𝑚+2 in terms of 𝑥.
𝟐𝟓𝒙
?
𝑚−𝑛
(b) Write 5
in terms of 𝑥 and 𝑦.
𝒙
𝒚
?
(c) Write 53𝑛 in terms of 𝑦.
𝒙𝟑 ?
(d) Write 5
𝑚+𝑛
2
in terms of 𝑥 and 𝑦.
𝒙𝒚 𝒐𝒓? 𝒙 𝒚
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
21 , 23 and 22 so that we have a?consistent base.
Skill 3: Changing bases
1
𝑥
10
Solve 4 = 2
2
Solve 2𝑥 =
22 𝑥 ?= 210
? 210
22𝑥 =
𝑥=
? 5
3
17
𝑥= ?
2
If 2 2 = 2𝑘 , determine 𝑘.
𝟐 𝟐=
𝟑
𝒌=
𝟐
𝟐𝟏
×
?
𝟏
𝟐𝟐
=
𝟑
𝟐𝟐
83
2
Test Your Understanding
1
If 9 3 = 3𝑘 , find 𝑘.
𝟗 𝟑=
𝟓
𝒌=
𝟐
2
𝟑𝟐
×
?
𝟏
𝟑𝟐
=
Solve 3𝑥 = 92𝑥−1
𝟑𝒙
𝟐𝒙−𝟏
𝟐
𝟑
=
𝟑𝒙 = 𝟑𝟒𝒙−𝟐
𝒙 = 𝟒𝒙 − 𝟐
?
𝟐 = 𝟑𝒙
𝟐
𝒙=
𝟑
𝟓
𝟑𝟐
Exercises
Solve for 𝑥:
1
8 𝑥 = 29
𝑥=3 ?
2
5𝑥 = 5 5
3
𝑥= ?
2
4
6
8
4𝑥 = 4
2
2𝑥 + 2𝑥 = 219
5
27𝑥 =
6
42𝑥+1 = 82𝑥−1
3
NN
3
4
𝑥=7 ?
𝑥 = 18?
𝑥 = 10?
930
9 × 27 =
𝑥
5
𝑥= ?
2
3
12
𝑥= ?
17
One Final Difficult GCSE question
𝑥 = 2𝑝 , 𝑦 = 2𝑞
a) Express in terms of 𝑥 and/or 𝑦.
? 𝑥𝑦
i) 2𝑝+𝑞 = 2𝑝 2𝑞 =
ii) 22𝑞 = 2𝑞 2 ?= 𝑦 2
iii) 2𝑝−1 =
2𝑝
𝑥
=?
1
2
2
b) Given that:
𝑥𝑦 = 32
2𝑥𝑦 2 = 32
find the value of 𝑝 and 𝑞.
𝑝 = 6,
?
𝑞 = −1
?