Radicals
Introduction: We know that
3
2, 3, 5, 15 …. are irrational numbers. Similarly,
2, 3 5, 3 14 etc are numbers which are the cube roots of some positive rational numbers,
which cannot be written as the cubes of any rational number. Such numbers are called surds
or radicals.
In general, if a is any positive rational number and n (>1) is a positive integer and if a cannot
be written as the nth power of any rational number, then
In other words,
(i)
(ii)
n
n
a is called a radical of order n.
a is a radical of order n if
a is a positive rational number,
n (>1) is a positive integer, and
n
(iii)
a is not a rational number.
All radicals are rational numbers. We state some laws of radicals without proof:
n
1.
2.
n
n
a
a
an b
n
ab
n
3.
a na
, where a and b are positive rational numbers and n (>1) is a positive
n
b
b
integer.
Example 1: State whether
Solution:
Therefore,
45 =
9 5
9
5
3 5 which is not a rational number.
45 is a radical.
Example 2: State whether
Solution:
45 is a radical or not with reason.
20
45 =
20
20 45
45 is a radical or not with reason.
900 = 30, which is a rational number.
Therefore,
20
45 is not a radical.
Practice problems:
Problem 1: State whether
3
4
3
54 is a radical or not with reason.
Problem 2: State whether 3 12 6 27 is a radical or not with reason.