Download Foundation Student Book Chapter 9 - Algebra

9 ALGEBRA 2
1.1 Section title
W
LO
S
E
R
C
I
P
In this chapter, we will look at writing very large numbers using powers. For example one
billion or one million million is written in long form as 1 000 000 000 000 but can be written
using powers as 1012. This form of writing large numbers is very useful in science. The
distance from the Earth to the Sun and back is approximately 108 miles.
162
Objectives
Before you start
In this chapter you will:
calculate with powers
write expressions as a single power of the same
number
use powers in algebra
multiply out brackets in algebra
factorise expressions.
You need to:
copy to follow…
9.1 Calculating with powers
9.1 Calculating with powers
Objectives
Why do this?
You can work out the value of numbers raised to
a power.
You can write numbers using index notation.
You can work out values of expressions given in
index notation.
Copy to follow?
Get Ready
1. Work out the following.
a 22
b 53
c 72
d 83
Key Points
The 2 in 52 is called a power or an index. It tells you how many times the given number must be multiplied by
itself. The plural of index is indices.
The 5 in 52 is called the base.
You can solve equations such as 3x  81 by working out how many times the base has to be multiplied by itself
to give the answer.
Example 1
Work out the following.
a 32
b 2 to the power of 5
a 32  3  3  9
3 to the power of 2 is usually called 3 squared.
b 2 to the power of 5  25
Write as a power.
25 means 2 multiplied by itself 5 times.
5
2  2  2  2  2  2  32
c 23  32
23  2  2  2  8
32  3  3  9
23  32  8  9
 72
Example 2
c 23  32
Work out 23 first.
Work out 32.
Use the two values above.
Rewrite these expressions using index notation.
a 3333
b 44555
c 66666
a 3  3  3  3  34
b 4  4  5  5  5  42  53
c 6  6  6  6  6 65
Replace 4  4 with 42 and 5  5  5 with 53.
The index is 5 because 5 lots of 6 are multiplied together.
power
index
indices
base
solve
163
Chapter 9 Algebra 2
Example 3
Find the value of x.
a 5x  25
b 4x  64
a 5  5  25
So 52  25
x2
Write as a power.
Compare powers.
b 4  4  4  64
Write as a power.
Compare powers.
So 43  64
x3
Exercise 9A
E
1
2
3
4
Questions in this chapter are targeted at the grades indicated.
Find the value of
a 25
d 10 to the power 4
b 4 to the power 4
e 54
Write these using index notation.
a 2222
c 111111
e 338888
Work out the value of
a 24
e 83
i 27  35
b 44444
d 888
f 4444222
b 35
f 24  93
j 43  41
c 63
g 26  45
Index
Value
3
10
1
10
5
164
6
Value in words
One thousand
100
2
1 000 000
D
d 52
h 53  34
Copy and complete the table for powers of 10.
Power of 10
5
c 16
f 6 to the power 5
One million
10
5
Work out the value of
a 43  102
d 102  52
b 4  102
e 103  23
c 6  103
f 43  22
Find x when
a 5x  125
e 9x  81
b 3x  81
f 3x  27
c 2x  64
g 2x  16
d 10x  10 000
h 7x  49
9.2 Writing expressions as a single power of the same number
9.2 Writing expressions as a single power of the
same number
Objective
Why do this?
You can use power rules to simplify
expressions.
Copy to follow?
Get Ready
1. Work out the value of
b 42  23
a 23  103
c 32  3
Key Points
To multiply powers of the same number, add the indices.
e.g. 32  33  32  3  35
To divide powers of the same number, subtract the indices.
e.g. 56  53  56 − 3  53
Any number raised to the power of 1 is equal to the number itself.
e.g. 41  4
To raise a power of a number to a further power, multiply the powers (or indices).
e.g. (103)2  103  2  106
Example 4
Simplify these expressions by writing them as a single power of the number.
a 23  24
b 58  53
c (82)5
a 23  24  23  4
Add the powers.
 27
b 58  53  58 – 3
Subtract the powers.
5
5
2 5
25
c (8 )  8
 810
(82)5  (8  8)  (8  8)  (8  8)  (8  8)  (8  8)
 810
Multiply the powers.
Exercise 9B
Simplify these expressions by writing as a single power of the number.
1
a 68  63
b 83  85
c 24  22
2
a 43  42
b 66  63
c 75  7
3
a 42  43
b 53  5
c 39  38
C
165
Chapter 9 Algebra 2
C
4
a 56  54  53
b 23  27  2
5
a 102  102  10
b 94  94
6
a 63  67  6
b 52  52  52
7
a 35  3  32
4
b 47  __
46
54
b 58  __
57
b (74)2
5
68  63
a __
62
a (52)3
8
9
49  45
c __
42
9.3 Using powers in algebra to simplify
expressions
Objectives
Why do this?
You can multiply powers of the same letter.
You can divide powers of the same letter.
You can raise a power of a letter to a further
power.
Get Ready
1. Simplify
a 72  75
b 85  83
c (53)2
Key Points
In the expression xn, the number n is called the power or index.
xm  xn  xm  n
xm  xn  xm  n
(xm)n  xm  n
Any letter raised to the power of 1 is equal to the letter itself, e.g. x1  x.
Example 5
Simplify
a x5  x3
e 3x2  4x3
a x5  x3  x5  3
x
b y7  y4
f 10x6  5x3
c a2  a3  a5
g (3a2)4
d (x3)2
Add the powers.
8
Examiner’s Tip
b y y y
7
4
7–4
 y3
c a2  a3  a5  a2  3  5
 a10
166
Subtract the powers.
Show your working.
The mark is scored here.
9.3 Using powers in algebra to simplify expressions
d (x3)2  x 3  2
Multiply the powers.
 x6
e 3x2  4x3  3  4  x2  x 3
 12  x
 12x 5
Write the numbers and letters together.
23
Multiply the numbers 3 and 4.
Add the powers 2 and 3.
10
f 10x 6  5x 3  __
 x 6 x 3
5
2x
 2x 3
Divide the numbers 10 and 5.
63
Subtract the powers.
Both 3 and a2 are raised to the power of 4.
34  3  3  3  3  81
g (3a 2)4  34  (a 2)4
 81  a 2  4
81a 8
Multiply the powers.
Exercise 9C
Simplify the following.
C
1
a x8  x2
b y3  y8
c x9  x5
2
a a5  a3
b b3  b3
c d 7  d4
3
a p5  p2
b q12  q2
c t8  t4
4
a j9  j3
b k5  k 4
c n25  n23
5
a x5  x2  x2
b y2  y 4  y3
c z 3  z 5  z2
6
a 3x2  2x3
b 5y9  3y20
c 6z 8  4z 2
7
a 12p8  4p3
b 15q5  3q3
c 6r 5  3r 2
8
a (d 3)4
b (e5)2
c ( f 3)3
d (g7)9
9
a (g6)4
b (h2)2
c (k 4)0
d (m0)56
10
a (3d 2)7
b (4e)3
c (3 f 129)0
11
a
a a4  __
a9
b
b b7  __
b4
c  c5
c c 3  __
c2
12
a 4d 9  2d
b 8e8  4e4
c (4 f 2)2
5
4
167
Chapter 9 Algebra 2
9.4 Understanding order of operations
Objective
Why do this?
You can work out the value of numerical
expressions.
When making a cake, you need to know what order
to add the ingredients in. The same is true of a
calculation such as 3  4  2  5. It is important
that the operations are carried out in the correct
order or the answer will be wrong.
Get Ready
1. Work out
a 63
b 42
c 70  7
Key Points
BIDMAS gives the order in which operations should be carried out.
Remember that B
I
D
M
A
S stands for
B rackets
If there are brackets, work out the value of the expression inside the brackets first.
I ndices
Indices include square roots, cube roots and powers.
D ivide
If there are no brackets, do dividing and multiplying before adding and subtracting, no
matter where they come in the expression.
M ultiply
A dd
S ubtract
Example 6
If an expression has only adding and subtracting then work it out from left to right.
Work out (3  2)  1
(3  2)  1  6  1
5
Example 7
Work out the Brackets first.
Work out 3  2  5  1
3  2  5  1  3  10  1
 13  1
 12
Example 8
Work out (10  2)2  5  32
(10  2)2  5  32  122  5  32
 144  5  9
 144  45
 99
168
operations
There is no Bracket or Divide, so start
with Multiply, then Add, then Subtract.
brackets
Brackets first, then Indices,
Multiply, and finally Subtract.
9.5 Multiplying out brackets in algebra
Exercise 9D
1
Use BIDMAS to help you find the value of these expressions.
a 5  (3  1)
b 5  (3  1)
c 5  (2  3)
d 523
e 3  (4  3)
f 543
g 20  4  1
h 20  (4  1)
i 642
j (6  4)  2
k 24  (6  2)
l 24  6  2
m 7  (4  2)
n 742
o ((15  5)  4)  ((2  3)  2)
2
Make these expressions correct by replacing the • with  or  or  or  and using brackets if you
need to. The first one is done for you.
a 4 • 5  9 becomes 4  5  9
b 4 • 5  20
c 2 • 3 • 4  20
d 3•2•55
e 5•2•39
f 4 • 2 • 8  10
g 5 • 4 • 5 • 2 27
h 5 • 4 • 5 • 2 23
3
Work out:
a (3  4)2
c 3  (4  5)2
e 2  (4  2)2
E
b 32  42
d 3  42  3  52
f 23  32
(2  5)2
h _______
32  2
j 42  24
g 2  (32  2)
52  22
i ______
3
k 25  52
l 43  82
9.5 Multiplying out brackets in algebra
Objectives
Why do this?
You can add expressions with brackets.
You can subtract expressions with brackets.
Copy to follow?
Get Ready
1. Expand
a 5(y  3)
b 3(4r  5s)
c 5x(2x  2y)
Key Points
Expanding brackets means multiplying each term inside the brackets by the term outside the brackets.
To simplify an expression with brackets, expand the brackets and collect like terms.
169
Chapter 9 Algebra 2
Example 9
Simplify
a 2(3x  y)  5(y  2x)
b 3x(2x  y)  2x(5y  1)
2 times (3x  y) plus 5
times (y  2x)
Multiply each pair of terms.
a 2(3x  y)  5( y  2x)
 2  (3x  y)  5  ( y  2x)
Remember   
  
  
  
 2  3x  2  y  5  y  5  2x
 6x  2y  5y  10x
 3y  4x
Collect the terms.
b 3x(2x  y)  2x(5y  1)
 3x  2x  3x  y  2x  5y  2x  1
 6x2  3xy  10xy  2x
 6x2  7xy  2x
x  x  x2
Collect the terms.
Exercise 9E
Expand and simplify.
C
170
1
3(x  2)  2(x  4)
2
4(2x  1)  3(4x  7)
3
5(3x  2)  4(2x  1)
4
7(3  2x)  3(2x  3)
5
6(4  2x)  3(5  3x)
6
4(3  2x)  3(1  5x)
7
2(3x  5y)  3(2x  4y)
8
5(6y  2x)  4(3x  2y)
9
3(2x  3y)  2(5x  6y)
10
3(2x  3y)  5(x  y)
11
4(3y  2)  5(y  2)
12
2(3x  6)  3(2x  5)
13
4(3  2x)  3(5  3x)
14
2(3  y  2x)  3(4x  3y)
15
3(2x  3y)  5(3x  2y)
16
5(3y  5x)  2(x  3y)
17
(4x  3y)  2(3x  2y)
18
7(3x  5y)  (x  3y)
19
x(2y  1)  2x(3y  1)
20
2x(3y  1)  y(2x  1)
21
2y(3x  2)  3x(2  3y)
22
4x(2y  5x)  2y(x  y)
9 .6 Factorising expressions
9.6 Factorising expressions
Objectives
Why do this?
You can factorise expressions by taking out a
single factor.
You can factorise an expression by taking out
multiple factors.
Copy to follow?
Get Ready
1. Find the common factors of these.
a 16 and 24
b 48 and 20
2. Factorise
a 2x  6
b 4y  12
c ab and abc
c 12p  40
d x2  2x
Key Points
Factorising is the reverse process to expanding brackets.
To factorise an expression, find the common factor of the terms in the expression and write the common factor
outside a bracket. Then complete the bracket with an expression which, when multiplied by the common factor,
gives the original expression.
Example 10
Factorise
a 3x2  5x  x(3x  5)
b 10a2  15ab
 5a(2a  3b)
a 3x2  5x
b 10a2  15ab
Take x outside the bracket. It is
a common factor of 3x2  5x.
The common factor may be both
a number and a letter.
Take 5a outside the bracket.
It is a common factor of both
10a2 and 15ab.
Examiner’s Tip
Always check your answer by
expanding.
Exercise 9F
Factorise each of the expressions in questions 1–6.
1
a 2x  6
d 4r  2
g 12x  16
b 6y  2
e 3x  5xy
h 9  3x
c 15b  5
f 12x  8y
i 9  15g
D
2
a 3x2  4x
d 5b2  2b
g 6m2  m
b 5y2  3y
e 7c  3c2
h 4xy  3x
c 2a2  a
f d 2  3d
i n3  8n2
C
171
Chapter 9 Algebra 2
C
3
a 8x2  4x
d 3b2  9b
g 21x4  14x3
b 6p2  3p
e 12a  3a2
h 16y3  12y2
c 6x2  3x
f 15c  10c 2
i 6d 4  4d 2
4
a ax2  ax
d qr2  q2
g 6a3  9a2
b pr2  pr
e a2x  ax2
h 8x3  4x4
c ab2  ab
f b2y  by2
i 18x3  12x5
5
a 12a2b  18ab2
d 4x2y  6xy2  2xy
b 4x2y  2xy2
e 12ax2  6a2x  3ax
c 4a2b  8ab2  12ab
f a2bc  ab2c  abc2
6
a 5x  20
d 4y  3y2
g cy2  cy
b 12y  10
e 8a  6a2
h 3dx2  6dx
c 3x2  5x
f 12b2  8b
i 9c 2d  15cd 2
Chapter review
The 2 in 52 is called a power or an index. It tells you how many times the give number must be multiplied by
itself. The plural of index is indices.
The 5 in 52 is called the base.
You can solve equations such as 3x  81 by working out how many times the base has to be multiplied by
itself to give the answer.
To multiply powers of the same number, add the indices.
To divide powers of the same number, subtract the indices.
Any number raised to the power of 1 is equal to the number itself.
To raise a power of a number to a further power, multiply the powers (or indices).
In the expression xn, the number n is called the power or index.
xm  xn  xm  n
xm  xn  xm  n
(xm)n  xm  n
Any letter raised to the power of 1 is equal to the letter itself, e.g. x1  x.
BIDMAS gives the order in which operations should be carried out.
Remember that B
D
M
A
S stands for
B rackets
If there are brackets, work out the value of the expression inside the brackets first.
I ndices
Indices include square roots, cube roots and powers.
D ivide
If there are no brackets, do dividing and multiplying before adding and subtracting, no
matter where they come in the expression.
M ultiply
A dd
S ubtract
172
I
If an expression has only adding and subtracting then work it out from left to right.
Chapter review
Expanding brackets means multiplying each term inside the brackets by the term outside the brackets.
To simplify an expression with brackets, expand the brackets and collect like terms.
Factorising is the reverse process to expanding brackets.
To factorise an expression, find the common factor of the terms in the expression and write the common
factor outside a bracket. Then complete the bracket with an expression which, when multiplied by the
common factor, gives the original expression.
Review exercise
1
2
3
4
5
6
7
Write down these using index notation.
a 666
b 11  11
E
c 222222
Write down the value of
a 54
b 27
c 103
d 105
Work out the value of
a 32  42
b 24  72
c 4  102
d 3  104
Rewrite these expressions using index notation.
a 22333
b 5577
c 448888
Work out
a 83
b 104
c 53
Find x when
a 3x  243
b 2x  32
c 10x  1000
Factorise
a 5x  15y
b 15p  9q
c cd  ce
d 24  32
Jake thinks of a number, squares it, multiplies his answer by 2 and gets 72.
What number did Jake think of?
9
Work out 65  (24  35).
10
In this set of squares, each number in a square is obtained by
multiplying the two numbers immediately underneath.
What number should go in the top square?
Give your answer as a power of 2.
12
13
e 52  25.
D
8
11
d 666222
AO2
2
4
4
2
C
Simplify
a 23  24
b 53  52
c 3  34
e 98  94
f 83  8
7
g 72  __
73
4
6
h 64  __
62
Simplify
a x6  x3
e x  x4
b x8  x5
f (x6)2
c (x3)5
g x8  x8
d x5  x4
h x7  x
i x2  x6  x3
j x8  x
x
k x6  __
x7
x  x5
l x3  __
x4
Simplify.
a 4x3  x5
e 24x6  3x
b 3x2  5x6
f 36x9  4x8
c 7x  3x4
g (x5)2
d 8x9  2x5
h (x3)3
d 75  72
4
7
173
Chapter 8 Algebra 2
C
14
15
16
AO2
174
17
Simplify
a a3  a4
b 3x2y  5xy3
Expand and simplify
a 3a(b  2a)  2b(3a  2b)
c 5c(3c  2d)  2c(c  d)
e 3a(b  c)  2b(a  c)  c(2a  3b)
Factorise
a x2  7x
b t2  at
c bx2  x
a Factorise x2  5x.
b Work out the value of 1052  5  105.
b 4p(2q  3p)  3p(2p  q)
d a(a  b)  b(a  b)
f 2a(b  2c)  3b(2a  3c)
d 3p2  py
e aq2  at
November 2007 adapted