Understanding Engineering Mathematics

am

an

=am−n

(am)n=amn

(ab)n=anbn

Note that for any indexn,1n=1.

Examples
23 × 24 = 23 +^4 = 27
35 × 30 = 35
( 52 )^3 = 56
( 22 )^0 = 20 = 1
43 / 22 =( 22 )^3 / 22 = 26 / 22 = 24

Asquare rootof a positive numbera, is any number that, when squared, yields the
numbera.Weuse

√
ato denote the positive value of the square root (although the notation
has to be stretched when we get to complex numbers). For example

2 =

√

4since2^2 = 4

Since− 2 =−

√

4 also satisfies(− 2 )^2 =4,−

√
4 is also a square root of 4. So the square
roots of 4 are±

√
4 =±2.
We can similarly have cube roots of a numbera, which yieldawhen they are cubed.
Ifais positive then^3

√

adenotes the positive value of the cube root. For example

2 =^3

√

8 because 2^3 = 8

In the case of taking an odd root of a negative number the convention is to let

√

denote the negative root value, as in^3

√

− 8 =−2, for example.

The correspondingnth root of a numberais denoted in general by

√na(also called aradical)

Ifnis even thenamust be positive to yield a real root (

√
−1isanimaginary number,
forming the basis of complex numbers, see Chapter 12). In this case, because(− 1 )^2 =1,
there will be at least two values for the root differing only by sign. Ifnis odd then the
nth root n

√

aexists for both positive and negative values ofa,asin^3

√

− 8 =−2 above.

Ifais a prime number such as 2, then

√
a is an irrational number, i.e. it can’t be
expressed in rational form as a ratio of integers (6
➤
). This is not just a mathematical
nicety.

√
2 for example, is the diagonal of the unit square, and yet because it is irrational,
it can never be written down exactly as a rational number or fraction (

√
2 = 1 .4142 is,
for example, only an approximation to

√

2 to four decimal places).

In terms of indices, roots are represented by fractional indices, for example:

√

a=a

1

2