Chapter 2. Exponents and Radicals

MODULE - 1
Algebra
Exponents and Radicals
2. State which of the following are surds:
Notes
(i) 3 64 (ii) 4 625 (iii) 6 216
(iv) 5 × 45 (v) 3 2 × 5 6
3. Identify pure and mixed surds out of the following:
(i) 32 (ii) 2 3
12
(iii) 13
3
91 (iv) 35
2.11 LAWS OF SURDS
Recall that the surds can be expressed as numbers with fractional exponents. Therefore,
laws of indices studied in this lesson before, are applicable to them also. Let us recall them
here:
(i) n
n n
x .
n
y =
n
xy or x .y = ( xy)n
1
1
1
(ii)
n
n
x
y
=
n
x
y
or
x
y
1
n
1
n
⎛ x ⎞
= ⎜ ⎟
⎝ y ⎠
1
n
(iii)
m n
x
=
mn
x
=
n m
x
or
⎛
⎜
x
⎝
1
n
⎞
⎟
⎠
1
m
= x
1
mn
⎛
= ⎜
x
⎝
1
m
⎞
⎟
⎠
1
n
m
1
x
n
= x or x n =
(iv) n m
m n
( )
(v) n p mn pn
p
pn
( ) ( ) mn
x
=
x
x
1
m
m
p
m
pn
mn
or x = x = x =
Here, x and y are positive rational numbers and m, n and p are positive integers.
Let us illustrate these laws by examples:
1 1
3 1
3 8 = 3 × 8 = 24 3 = 24 = 3×
8
(i) 3 3 3
3 3
( )
x
1
(ii)
() 5
() 9
1
3
1
3
⎛ 5 ⎞
= ⎜ ⎟
⎝ 9 ⎠
1
3
=
3
5
9
58
Mathematics Secondary Course