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Chapter 9

Basic algebra
9
...

For example, if the length of a football pitch is L and its
width is b, then the formula for the area A is given by
A= L ×b
This is an algebraic equation
...

The total resistance, RT , of resistors R1 , R2 and R3
connected in series is given by
RT = R1 + R2 + R3
This is an algebraic equation
...
3 k, R2 = 2
...
5 k, then
RT = 6
...
4 + 8
...
2 k
The temperature in Fahrenheit, F, is given by
9
F = C + 32
5
where C is the temperature in Celsius
...

9
If C = 100◦C, then F = × 100 + 32
5
= 180 + 32 = 212◦F
...


9
...

If, say, a, b, c and d represent any four numbers then in
algebra:
DOI: 10
...
00009-0

(a) a + a + a + a = 4a
...

(b) 5b means 5 × b
...

(c) 2a + 3b + a − 2b = 2a + a + 3b − 2b = 3a + b
Only similar terms can be combined in algebra
...

In addition, with terms separated by + and − signs,
the order in which they are written does not matter
...
(Note that the first term, i
...

2a, means +2a
...
(Note
that − × − = +)
(e) (a)(c)(d) means a × c × d
Brackets are often used instead of multiplication
signs
...

(f ) ab = ba
If a = 2 and b = 3 then 2 × 3 is exactly the same
as 3 × 2, i
...
6
...
For
32 = 3 × 3 = 9
...

Here are some worked examples to help get a feel for
basic operations in this introduction to algebra
...
2
...


Find the sum of 4x, 3x, −2x and −x

4x + 3x + −2x + −x = 4x + 3x − 2x − x
(Note that + ×− = −)
= 4x
Problem 2
...


Simplify 3 + x + 5x − 2 − 4x
...


Add x − 2y + 3 to 3x + 4y − 1
...


Subtract a − 2b from 4a + 3b
...


From a + b − 2c take 3a + 2b − 4c
...
From x 2 + x y − y 2 take x y − 2x 2
...
2
...


Simplify bc × abc

5x+3y + z + −3x + −4y + 6z
= 5x + 3y + z − 3x − 4y + 6z

bc × abc = a × b × b × c × c

= 5x − 3x + 3y − 4y + z + 6z

= a × b 2 × c2

= 2x − y + 7z

= ab2 c 2

Note that the order can be changed when terms are separated by + and − signs
...


Problem 6
...


Simplify 4x 2 − x

− 2y + 5x + 3y
Problem 7
...


ab × b2c × a = a × a × b × b × b × c
= a 2 × b3 × c

Simplify 3x y − 7x + 4x y + 2x

3x y − 7x + 4x y + 2x = 3x y + 4x y + 2x − 7x
= 7xy − 5x

Simplify ab × b2c × a

= a2 b3 c
Problem 8
...


Find the sum of 4a, −2a, 3a and −8a
...


Find the sum of 2a, 5b, −3c, −a, −3b and
7c
...


Simplify 2x − 3x 2 − 7y + x + 4y − 2y 2
...


Simplify 5ab − 4a + ab + a
...


Simplify 2x − 3y + 5z − x − 2y + 3z + 5x
...


5 pq 2r 3 = 5 × p × q 2 × r 3
 2  3
1
2
= 5×2×
× 2
5
2

Basic algebra
 2  3
2
5
= 5×2×
×
5
2

1 5
since 2 =
2 2

5 2 2 2 5 5 5
× × × × × ×
1 1 5 5 2 2 2
1 1 1 1 1 5 5
= × × × × × ×
1 1 1 1 1 1 1
= 5×5

Problem 13
...
Multiply 2a + 3b by a + b
Each term in the first expression is multiplied by a, then
each term in the first expression is multiplied by b and
the two results are added
...

2a + 3b
a+b
Multiplying by a gives
Multiplying by b gives

2a 2 + 3ab
2ab + 3b2

Adding gives

2a 2 + 5ab + 3b2

Thus, (2a + 3b)(a + b) = 2a2 + 5ab + 3b2
Problem 11
...
The usual layout is shown below with the
dividend and divisor both arranged in descending
powers of the symbols
...


...
e
...

(iii) The divisor is then multiplied by 2x, i
...

2x(x − 1) = 2x 2 − 2x, which is placed under the
dividend as shown
...

(iv) The process is then repeated, i
...
the first term
of the divisor, x, is divided into 3x, giving +3,
which is placed above the dividend as shown
...
The remainder, on subtraction, is zero,
which completes the process
...
Simplify 2x ÷ 8x y
2x
8x y
2x
2×x
=
8x y 8 × x × y
1×1
=
4×1× y
1
=
4y

Problem 14
...


2x ÷ 8x y means

(A check can be made on this answer by
multiplying (2x + 3) by (x − 1), which equals
2x 2 + x − 3
...
Simplify

x 3 + y3
x+y

64 Basic Engineering Mathematics
(i) (iv) (vii)

Alternatively, the answer may be expressed as

2
2
x − xy + y
3
x + y x + 0 + 0 + y3
x3 + x2 y

−x 2 y

4b 3
4a 3 − 6a 2 b + 5b3
= 2a 2 − 2ab − b2 +
2a − b
2a − b

+ y3

Now try the following Practice Exercise

−x 2 y − x y 2

Practice Exercise 36 Basic operations in
algebra (answers on page 343)

x y2 + y3
x y2 + y3

...


1
...


2
...


(i)

x into x 3 goes x 2
...


3
...


(ii)

x 2 (x + y) = x 3 + x 2 y

4
...


5
...

(iv)

x into −x y goes −x y
...

2

(v) −x y(x + y) = −x 2 y − x y 2

6
...

(vii)

x into x y 2 goes y 2
...


(viii)

y 2 (x + y) = x y 2 + y 3

(ix) Subtract
...

Thus,
x+y
The zeros shown in the dividend are not normally shown,
but are included to clarify the subtraction process and
to keep similar terms in their respective columns
...


Divide 4a 3 − 6a 2 b + 5b3
2a 2 −

by 2a − b

8
...


9
...


25x 2 y z 3
5x yz

10
...

11
...

12
...

13
...

14
...


2ab − b2


2a − b 4a 3 − 6a 2 b
4a 3 − 2a2 b

+ 5b3

−4a 2 b

+ 5b3

15
...

16
...

17
...


−4a 2 b + 2ab2
−2ab2 + 5b3
−2ab2 + b3
4b3
4a 3 − 6a 2 b + 5b3
= 2a 2 − 2ab − b2 , remain2a − b
der 4b3
...
3

Laws of indices

The laws of indices with numbers were covered in
Chapter 7; the laws of indices in algebraic terms are
as follows:
(1) am ×an = am + n

For example, a 3 × a 4 = a 3+4 = a 7

Basic algebra
(2)

am
= am−n
an

(3) (am )n = amn
√
n

m

(4) a n =
(5) a−n =

am

1
an

(6) a0 = 1

c5
= c5−2 = c3
c2
 3
For example, d 2 = d 2×3 = d 6

For example,

4

For example, x 3 =

√
3

For example, 3−2 =

x4

1
1
=
32
9

a 3 b 2 c4
Problem 20
...

Problem 17
...
Simplify ( p3 )2 (q 2 )4

a 2 × a 1 × b 3 × b 2 × c1 × c5
Using law (3) of indices gives

Using law (1) of indices gives
a 2+1 × b3+2 × c1+5 = a 3 × b5 × c6
a 2 b3c × ab2 c 5 = a 3 b5 c 6

i
...


1

3

1

1

Problem 18
...
Simplify

Problem 22
...
Using law (3) of indices
gives
(mn 2 )3
m 1×3 n 2×3
m3n6
=
=
(m 1/2 n 1/4)4
m (1/2)×4n (1/4)×4 m 2 n 1
m3 n6
= m 3−2 n 6−1 = mn5
m2 n1
Problem 23
...
Simplify

66 Basic Engineering Mathematics
Using law (3) of indices gives

p1/2 q 2r 2/3
and evaluate
p1/4 q 1/2r 1/6
when p = 16, q = 9 and r = 4, taking positive roots
only

Problem 25
...

1
...


a × a2 × a5

3
...


b4 × b7

5
...


c5 × c3 ÷ c4
(x 2 )(x)
x6
 2 −3
y
 −7 −2
c
 4
1
b3
1
 3
s3

9
...

13
...


17
...

12
...

16
...


x 3 y2 z
x 5 y z3

19
...

2
a 5 bc3

and
20
...


Simplify

when

Here are some further worked examples on the laws of
indices

x 2 y3 + x y2
xy
a +b
can be split
c

Algebraic expressions of the form
a b
into +
...

Problem 27
...

Dividing each term by x y gives

x2 y
evaluate

1

= (2)(33 )(2) = 108

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