# 1.2 Indices and Surds

```If a is a real number and n is a positive
integer, then
a
base
n
= a.a.a.a … a (n times)
index
The number a is called the base and n is
called the index, and a n is read as
‘ a to the power of n ‘
1
RULES OF INDICES
1. a  a  a
m
n
2. a  a  a
m
3. ( a
n
m
) a
n
m n
mn
mn
Notes
1.
2.
3.
a
a
0
n
1
n
 1
1
 n
a
a  a
m
n
 a
m
a b
n

n
4.
a
5.
ab
6.
an a
   n
b
b
n
n
n
n
3
Example 1
Evaluate each of the following without using
calculator.
(a)
8
2
3
 8 
(b) 

 27 
-
2
3
4
Example 2
Simplify each of the following.
3
2
(a)
(y ) y
(c)
x y
2
2
x y
2
3
(b)
( t ) ( t )
3
5
2
5
Example 3
Simplify
4 6
3
form 2
m
n
3
8
1
2
in the
12
and 3
6
SURDS
OBJECTIVES
(a)Explain the meaning of a surd and its
conjugate, and to carry out algebraic
operations on surds
(b) State the rules of indices
Surds
An irrational number and
expressed in terms of root sign
Positive
integer
n
a
Real
number
Note: a is not a perfect square, a > 0
Let’s pronounce correctly
a
is nth root of a
a
is square root of a
3
a
is cube root of a
4
a
is fourth root of a
n
9
Rules of surds
1
n
1.
n
a a
2.
n
a a
3.
n
ab  a b
n
a na
n
b
b
4.
5.
n
n
m n
n
a 
mn
a
10
Algebraic Operations on Surds
a) Multiplication
a× a
=a
a× b
= ab
Example 1:
When unlike surds
are multiplied
together, the
product is a surd.
a)
3 × 3  3 3 =3
b)
5 × 2  5 2 = 10
b) Division
a =1
a÷ a =
a
a
a÷ b =
Example 2:
a)
2÷ 2
b) 21 ÷ 5
=
b
2
2
21
=
5
=
a
b
=1
21
=
5
c) Addition and Subtraction
a c ±b c =  a ± b  c
Example 3:
i) 4 3 +2 3
=  4 + 2 3
=6 3
ii) 4 3 -2 3
=  4 -2 3
=2 3
d) Expansion of Surds

a± b
 =
2
a± b

a± b

= a a± a b± b a± b b
= a ± b ± 2 ab
Expansion of Surds (Alternatively) :

a± b
 = a
2
2
±2 a b +
= a ±2 ab +b
 b
2

a± b
Example 4:
i)

2+ 3

2

2
=
=

a
 2

2
2
±2 a b +
+2 2 3 +
= 2 + 3 + 2 2 3
=5 +2 6

b
 3
2

2
Rationalising the Denominator
Surd
Conjugate
2 + 3 
-4 - 5 
The conjugate of the surd
a+ b
is
a- b
Example 5 :
Rationalise the denominators of each of
the following fractions.
8
8
2
8
2
(a)
=4 2
=
×
=
2
2
2
2
Solving Equations Involving Surds
Example 6:
Solve the following equations. Give your
answer in the set form.
(a) 6x +1 -5 = 0
(b) 2 x  4 - x -1  4
```