Understanding Engineering Mathematics

and in general

√na=an^1

This fits in with the rules of indices, since

(

a

1

n

)n

=a

1

n×n=a^1 =a

Fractional powers satisfy the same rules of indices as integer powers – but there are
some new features:

• multiplicity of roots: 2^2 =(− 2 )^2 = 4


• non-existence of certain roots of negative numbers:

√

−1 is not a real

number

• irrational values for roots of primes and their multiples:

√

2 cannot be

expressed as a fraction

Quantities such as

√

2 ,

√
3 ,...containing square roots of primes, are calledsurds.The
term originates from the Greek word for mute, referring to a number that cannot ‘speak’ its
value – because its decimal part never ends (see Section 1.2.8). In mathematical manipu-
lation surds are always best left as they are – retaining the root sign. Any decimal form for
them will simply be an approximation as noted for

√
2 above. Usually we try to manipulate
surds so that the result is the simplest form, and none remain in denominators (although

we would normally write, for example, sin 45°=

1

√

2

). To do this we can use the rules of

indices, and also a process known asrationalisation, in which surds in denominators are
moved to the numerator. The ideas are illustrated in the solution to the review question.

Solution to review question 1.1.7

A. (i) 2^324 = 23 +^4 = 27 (leave it as a power, like this)

(ii)

34

33

= 34 −^3 = 31 = 3

(iii)( 52 )^3 = 52 ×^3 = 56

(iv)

( 3 × 4 )^4

( 9 × 23 )

=

3444

9 × 23

(note both 3 and 4 are raised to the power 4)

Note that 9= 32 ,4= 22 , so we can write,

=

34 ( 22 )^4

3223

=

3428

3223

= 34 −^228 −^3

= 3225

(v)

162

44

=

( 42 )^2

44

=

44

44

= 1