example.1
Arrange
the following terms in descending order: 
and 
Solution
Whenever the exponents are fractions, the
least natural number divisible by the denominators of the exponents is to be
found. .
The least common multiple for 4, 6 and 3 is
12. .
31/4 = (33)1/12
51/6 = (52)1/12
21/3 = (24)1/12
The descending order is 27, 25, 16. .
Hence 31/4, 51/6 and 21/3
are in descending order. .
example.2
Simplify: 
.
Solution
Rewriting the above expression with a common
denominator,
= 
= 
example.3
Find the square root of 37 + 12
.
Solution
The square root of a number of the form p + q
where p, q and r are rational numbers will be of the form 
+ 
where s and t are real numbers. In this problem, p = 37, q = 12
and r = 7
Hence 
Squaring both sides, 37 + 12= s + t + 2
equating the rational and irrational parts on both sides, s + t =
37 → (1)
i.e., 
&⇒ 4st = 1008
(s – t) = 
which is greater than zero as s > t
As s + t = 37 and 4st = 1008,
s – t = 
s – t = 
= 19 → (2)
Adding (1) and (2)
2s = 56
s = 28
t = s – 19 = 9
Hence 
example.4
Compare
and find as to which of the following surds is greater:
and 
Solution
Squaring and, we get 7 + 19 + 2
and 3 + 29 + 2
i.e., 26 + 2and 32 + 2
respectively.
26 + 2= 26 + 
= 26 + (23, 24) = (49, 50)
\ 26 + 2lies between 49 and 50
32 + 2 = 32 + 
= 32 + (18, 19) = (50, 51)
\ 32 + 2
lies between 50 and 51.
&⇒ 32 + 2> (26 + 2)
example.5
Arrange
the numbers below in ascending order and identify the correct choice. .
(a) 
(b) 363
(c) 3333
(d) (33)3
(1) cdba
(2) dabc
(3) cadb
(4) dbca
Solution
= 327 As 33 lies between 27(33) and 81(34),
3333 lies between (33)33 and (34)33
i.e., 399 and 3132
(33)3 = 39
(33)3 < (327)
< 363 < 3333
Hence dabc represents the ascending order. .
