Study Advice Service
Student Support Services
Mathematics
Worksheet
Algebra 1
This is one of a series of worksheets designed to help you increase your confidence in
handling Mathematics. This worksheet contains both theory and exercises which cover:1.
2.
3.
4.
Using letters for number
Substitution
Simplification of expressions
Indices
There are often different ways of doing things in Mathematics and the methods
suggested in the worksheets may not be the ones you were taught. If you are successful
and happy with the methods you use it may not be necessary for you to change them. If
you have problems or need help in any part of the work then there are a number of ways
you can get help.
For students at the University of Hull
 Ask your lecturers



You can contact a Mathematics Tutor from the Study Advice Service on the
ground floor of the Brynmor Jones Library if you are in Hull or in the Keith
Donaldson Library if you are in Scarborough; you can also contact us by email
Come to a Drop-In session organised for your department
Look at one of the many textbooks in the library.
For others
 Ask your lecturers
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 Use any other facilities that may be available.
If you do find anything you may think is incorrect (in the text or answers) or want further
help please contact us by email.
Tel:
01482 466199
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Email: studyadvice@hull.ac.uk
1. Using Letters for Numbers
In arithmetic we often use shorthand to simplify what we write. For instance we write
3  3  3  3 as 3  4 . In algebra we use the same sort of shorthand and write as
b  4 which we simplify further and write as 4b . When we multiply two numbers it
doesn’t matter which order we do it in: 6  5  5  6  30 , so 4b can be thought of as
4  b or b  4 .
This rule is called the commutative law of multiplication. Notice that addition is also
commutative 3  4  4  3  7 but subtraction and division are not.
( 3  5  5  3 and 6  3  3  6 ).
If b  3 then 4b  4  3  3  4  12 .
If b  26.5 then 4b  4  26.5  26.5  4  106 .
4b is shorthand for all possible products of 4 and some other number represented
by the letter b .
In the same way we use xy to represent x  y . Also we can see that the products
x  y and y  x are the same, so xy  yx .
It is very important to realise that x  y and xy are not the same.
If we have two pieces of wood whose lengths, in metres, are x and y then the total
length of the two pieces of wood laid end to end is x  y metres and the area of the
rectangle of sides x metres and y metres is xy square metres.
This can be seen in the diagram:
x
length x + y
x
y
y
Area xy
So x  y ( a length) is not the same as xy (an area).
If x  y , the rectangle becomes a square with area xx which we write as x 2 .
In the same way we write m  m  m as m3
We also write ( mn)2 to represent mn  mn which is m  n  m  n  m2n2
page1
2. Substitution
If you are given the value that each letter in an algebraic expression represents, you
can substitute those numbers into the expression and then, using normal arithmetic,
evaluate (find the value of) the expression.
If x  2, y  4, then
x  5 y  2  5  4  2  20  22
xy  2  4  8
3x 3  3  23  3  8  24
3x 3  3  23  63  6  6  6  216
5 x 3 y 2  5  23  42  5  8  16  640
In many areas of mathematics, engineering etc. particular letters and symbols are
used to stand for particular things. For instance the Greek letter  (pi) is used to
represent the number 3.14159… though, usually, you will be told to take  as 3.14,
3.142 or 22
.
7
x is standard shorthand for ‘mean value’.
The symbol
 means the ‘the sum of’ etc.
Formulas are a way of expressing connections between a number of variables in
mathematical shorthand.
If you travel for t hours at u km/h then s  ut gives the distance travelled s km.
s  4.9t 2 gives the approximate distance (s metres) an object falls under gravity in t
seconds and s  16t 2 gives the approximate distance in feet.

r
A  P 1  100
n gives the amount you owe (£A) after n months if you borrow £P at
r % interest per month (assuming you pay nothing back).
x
x
n
gives the mean value of n numbers.
Formulas and expressions occur in many different areas of life and it is important to
be able to use them by substituting values into them to obtain results.
Examples
Find the value of y in each of the following by substituting the given values
(a) y  3x  5 where x  2
y  3x  5  3  2  5  6  5  11
(b) y  x 2  2 x  3 where x  1.5
y  x 2  2 x  3  1.52  2 1.5  3  2.25  3  3  2.25
page2
(c) y 
y
27 z  5 x
xz 2  5
27 z  5 x
xz 2  5
where x  3, z  1

27  1  5 3
 3  12  5

27  15 42

 21
35
2
12
where x  4, z  0
2z  x
12
12
y  3 x 2  3 xz 
 342  3  4  0 
 48  0  3  51
2z  x
2 0  4
(d) y  3 x 2  3 xz 
Exercise 1
a  2, b  3, c  4, x  1 and y  5 find the values of
4. 2axy
3. 4abc
1. 2ax
2. 3xy
8. 4 x 2 a 2
7. 3x 3
5. 3axc
6. 2a 2
9. 2c 3
10. 3x 2 y 2
11. 2 y 2 c 2
12. 3x 3 y 2 c 2
14. 3a 2 b 2 c 3 y 2
15. 3xy2
16. 3xy2
13. 2 xa3 y 2
Exercise 2
A) Find the value of y in the following
1. y  3 x  7; x  6
2. y  x 2  7 x, x  2
3. y  4 x 2  2 x  2, x  3.5
4. y  1  3 x 2 x  3, x  8

5. y 
x  7x
 4 x, x  1
3x  2
2
6. y 

5x  7
4x 2  2
 6 x; x  0.5
B) Find the values of the following
(take  as 3.142 and give answers to 3 significant figures where appropriate)
2. s  ut  12 ft 2 , u  2, t  3, f  10
1. C  2 r ; r  10
l
; l  23, g  9.81
g
3. V  43 r 3 ; r  3.2
4. T  2
1 r 2 
; r  3, y  8
5. x  y
1 r 2 


 rty  1 
 ; r  27, t  2, y  3
6. q  
t  yr
3. Simplification of expressions
To add the numbers
75, 75, 76, 75, 74, 73, 75, 76, 75, 74, 76, 75, 75, 76, 75
you might count how many times each number occurs and simplify the working to
give the total as
(1 73)  (2  74)  (8  75)  (4  76)  1125 .
This is easier and safer than taking the terms one at a time and adding as you go
along.
page3
In an algebraic expression you can also collect like terms.
a  a  a  3a ;
3a  4a  2a  5a
a  b  b  c  a  b  c  2  a  3  b  2  c  2a  3b  2c
If terms have different letters (such as 3a, 4ab, 7 x ...) or the same letters to different
powers (such as a, a 2 , 2a 3 ) then you cannot put them together.
Consider the pieces of wood again. Before, we showed that x  y represents a
length in metres while xy represents an area in square metres.
You cannot combine a length in metres and an area in square metres, in the same
way you cannot write the expression x  y  xy in any simpler form.
If x  y then the length becomes 2 y metres and the area y 2 square metres. Again
you, obviously, cannot write 2 y  y 2 as 3 y 2 or 4 y . It is not possible to write the
expression 2 y  y 2 more simply. 2 y and y 2 are unlike terms
The terms x , y and xy are unlike terms and cannot be combined.
Pairs such as
a and 2a ;
x3 and 7x3 ; 4 x3 y and  3 x3 y are like terms.
Pairs such as
x3 and y 2 ; x3 and 3x5 ; 6x and 24; 3xy and x3 y are unlike
terms.
We can simplify expressions such as 3a 2  4a  c  16  2a 2  5c  12 by collecting like
terms together.
But remember the sign is part of the term and stays with it all the time!
3a 2  4a  c  16  2a 2  5c  12
Hence

3a 2  2a 2  4a  c  5c  16  12

a2
 4a
 4c
4
(Note we write 1a2 as a 2 )
To multiply and divide algebraic quantities we use the same rules as for numbers
(this is fairly obvious as the letters represent numbers!).
Just as as 3  4  5  3  5  4  5  3  4 etc.
We have  2x    3 y   2  x   3  y  2  (3)  x  y   6   xy   6 xy

 

And 5m2n  3m2n2   5  3   m2  m2  n  n2  15m4 n3
When dividing algebraic expressions, cancellation between numerator and
denominator is often possible. Cancelling is equivalent to dividing both the numerator
and denominator by the same quantity, just the same as dealing with equivalent
6  36  1
fractions ( 63  12
etc.).
72
2
page4
For example
4x  2 y 
4x
2x

2y
y
pq
pq
q


 q
p
p
1
s4
s
18 x 3 y 2
6 xy 2 z
3


(dividing top and bottom by 2)
(dividing top and bottom by p )
ssss
 s (writing s1 as s ).
sss
18  x  x  x  y  y
3x 2

6 x y  y  z
z
When the expression has a mixture of multiplication and division it is important to get
things in the correct order. Brackets first, then division & multiplication and finally
addition and subtraction!
In the following square brackets are used to show what is implied by some of the
multiplication signs:
(a) 2  3  5  6  [2  3]  [5  6]  6  30  36
(b) 2  (3  5)  6  2  8  6  16  6 or 2  48  96
(c) 2  (3  5  6)  2  (3  [5  6])  2  (3  30)  2  33  66
Examples
(a) Simplify 14 xy  5 yx  7 x  9 y  2 xy  3 y
Collecting like terms and remembering yx  xy , we get:
14 xy  5 yx  7 x  9 y  2 xy  3 y  14 xy  5 xy  2 xy  7 x  9 y  3 y
 21xy  7 x  6 y
(b) Simplify 3x   2a   4 x 
6x 2 a 3
 2a
2 xa 2
First do the multiplication and the division
6x 2 a 3
3x   2a   4 x 
 2a   6ax  4 x  3ax  2a
2 xa 2
then collect like terms giving  9ax  4 x  2a
page5
Exercise 3
Simplify where possible. If it is not possible say so
2. 2 p  2 12 p  3 p  p 2
1. 4s  2s  10s 2  8s
3. 3 x  2 y  4 x  3 y  8 z  x  6 y
4. 2abcd  3acbd  12dcba
5. xy  7 xy  3 x  5 y  2 xy
6. 3 x 2  2 x  x 2  x 4
7. 2 x  4 y  3 x  4 y  7 y
8. 10ab  13ab  2ba  7 ab
9. 3a 2 b  2a 3 b  5ab 3  4a 2 b
10.  6ab  5ba  a  b
11. 5 x  4 y  10 xy
2
13.
x4 y3
x2 y

12. 6 p 2q 4  25qrp 2  15 p 4
3y 5 x 2 z
14.
2y3z
3x 4 yz 3 25 x3 4

z
5 xz
yz
4. Indices
You will know or have noticed that
x 4  x 3  x 43  x 7 ; 5 4  5 6  5 46  510
and
y8  y3  y83  y5 ; 37  3 2  372  35
You may have met the ‘rule’ that to multiply expressions with the same base you
‘add’ the indices and to divide you ‘subtract’ the indices, as long as the index at the
top is greater than that at the bottom. These rules can be extended, by definition,
to cover all indices.
t3
t3
If we consider 5 the ‘rule’ gives
 t 3  t 5  t 35  t  2
5
t
t
3
t
t t t
1
expanding gives 5 
 2
t t t t t
t
t
1
So we want t 2 to be the same as 2 .
t
0
But what about t ? The ‘rule gives’
t6
t6
t
6 6
 t but we know that
0
t6
t6
 1
So we want t 0 to be 1.
We know that
t  t  t (for instance
that, if the ‘rule’ works, t
1
2
 t
1
2
9  9  3  3  9 etc.) and we can also see
 t1  t
1
So we want t 2  t
We extend the meaning and use of indices by defining
1
1
t  n as
, t n as n t and t 0 as 1 where t  0
tn
It is important to realise that these rules are defined to be true for all indices
(positive, negative or fractional).
page6
To summarise the rules:
For all values of m, n and a (as long as a is not 0)
(i) a m  a n  a m n
(ii) a m  a n  a m n
a 
m n
(iii)
m
Combining the last two means that a n  n a m 
1
 a mn
(iv) a n  n a
n a m
Examples
(a) Simplify 32 (without using a calculator!!)
1 1
32  2 
9
3
3
(b) Simplify 9 2 (without using a calculator!!)
3
9 2  2 93  729  27
3
or, better to use, 9 2  2 93 
 9
3
 33  27
5
(c) Simplify 16 4 (without using a calculator!!)
 
5
16 4 = 16 5
or
1
4
which is not easy to work out!
 
5
1 5
5
16 4 =  16 4   4 16  2 5  32


(d) Simplify
x 2  xy
, giving your answer in index form.
x
x 2  xy
x

x 2  xy
1
x2

x3 y
1
x
(3  12 )
5
y  x2 y
x2
Exercise 4 Without using a calculator, simplify the following, leaving your answers
as fractions where necessary.
1
1. 16 2
 4
6.  
5
2. 16
2
 12
3
92
7.  
 4
11. 35  3  5 12. 125 3
 2
3.  
 3
1
8.  
 3
1
13.
1
22
2
3
7
22
1
 9 2
4.  
 25 
1
72

9.  1 
 9
14.
5
53
2
53
5. 36 0
 1 
10.  
 27 
 32
 163 
15. 

 259 
0
page7
Exercise 5 Simplify
1. 3 x 2  2 x 5
2. 7 x 3  3x 4
3. 4 x 7  2 x 4  3x 3
4. y 12  y 4
5. 12a 7  3a 3
6. 28t 5  7t 7
7. 14a 7 b10  7 a 5 b 5
8. 10 x10  4 x 8  5 x 3
9. 42s 2  7 s 2  3s 3
10.
2 y 4 14 y 6

y5
y3
ANSWERS
Exercise 1
1 4
2 15
3 96
4 20
5 24
6 8
7 3
8 16
9 128
10 75
11 800
12 1200
13 400
14 172800
15 75
16 225
11. vm3 n  m 4 n 2
Exercise 2
A
1 11
2 -10
3 44
4 39
5 5.2
6 -1.5
B
1 62.8 (to 3 sig. figs)
2 51
3 137 (to 3 sig. figs)
4 9.62 (to 3 sig. figs)
5 10
6 2.71 (to 3 sig. figs)
3
1
12. 3 2  24 2  81
 12
 2
Exercise 3
1 10 s  10 s 2
2 1 12 p  p 2
3 y  8z
4  7 abcd
5 3 x  5 y  4 xy
6 x4  2 x2  2 x
7 5x  y
8 2ab
9 7 a 2b  2a 3b  5ab3
10 a  b  ab
11 2 y
12 10q 5r
13
5
2
x2 y2
14 15 x 6 z 5
page8
Exercise 4
1. 4
2.
3.
4.
1
4
4
9
3
5
5. 1
6.
7.
25
16
27
8
Exercise 5
8. 27
9.
10.
11.
12.
13.
4
3
 1 13
9
1
5
16
14. 5
15. 1
1. 6 x
7. 2 a 2 b 5
7
2. 21x 7
8. 8 x 15
3. 24 x14
9. 18s 3
4. y 8
10. 28 y 2
5. 4 a 4
11. vmn1 
6. 4t  2 
4
t
vm
n
12. 2
2
We would appreciate your comments on this worksheet,
especially if you’ve found any errors, so that we can improve it for
future use. Please contact the Maths tutor by email at
studyadvice@hull.ac.uk
Updated 24th November 2004
The information in this leaflet can be made available in an
alternative format on request. Telephone 01482 466199
page9