Understanding Engineering Mathematics

2.1.12 Powers and indices for algebraic expressions ➤^7078 ➤➤

Simplify the following

(i)

a^2 c^4

a−^3 b^2 c

(ii)

( 3 x)^3 ( 2 y)−^1

(x^2 y)−^1

(iii)

(x− 1 )

1

(^4) ( 16 x+ 16 )
1
4
(x+ 1 )(x^2 − 1 )−
3
4
2.1.13 The binomial theorem ➤^7179 ➤➤
Expand( 2 −x)^5 by the binomial theorem.

2.2 Revision

2.2.1 Multiplication of linear expressions

➤

38 73 ➤

Algebrais the branch of mathematics that generalises arithmetic by usingsymbolsas well
as numbers. Symbols, such asx,y,a,b, may represent any numbers, but there are certain
conventions used which are useful to know: letters from the beginning of the alphabet:
a,b,c,...usually representconstants– quantities having a fixed value; letters from the
middle of the alphabet...l,m,n,...usually representwhole numbersorintegers; letters
from the end of the alphabet...x,y,zusually representvariables, which may take a range
of values. As mathematics becomes more advanced we need more symbols and the Greek
alphabet is pressed into service – you have already seen an example inπ, the Greekp.
Analgebraic expressionis any quantity built up from such a finite number of symbols
using only the arithmetic operations ofaddition, subtraction, multiplicationordivision.
This includes integer powers which are simply successive multiplication, and roots of
variables, such as

√

x.

Examples

x+y,

a+ 2 b

a−b

,x^2 + 2 x+ 3 ,

√

t+

1

√

t− 1

are all algebraic expressions, in the symbols indicated. Be careful to distinguish between
analgebraic expression,suchasx^2 −1, and analgebraic equationsuch asx^2 − 1 =0.
An expression tells you nothing about the variables involved, it stands alone, whereas an
equation can fix the values of the variables.
Most of the expressions we will deal with in fact contain just one variable – called
functions of a single variable– which is traditionally, but not exclusively, denoted byx.
The simplest example is thelinear expression inxsuch as 2x+3 (called linear because
its graph is a straight line (➤212), or in general

ax+b

whereaandbare given constants. If the use of symbolsa,bto represent constants worries
you, just think of them as numbers such as 2, 3, etc.