So far, you read about introduction of Radical, Now, we shall know about other forms of radical or surd with solved examples: I hope you will find it easy and interesting.
A
surd or radical in the simplest form:-
(i)
It has no fraction under the radical sign,
(ii)
The radical has no factor with e×ponent n, and
(iii)
The given surd is not equal to any surd of
order lower than n
We shall consider on following e×ample to see this:-
(i)
3√2/3, (ii) 3√3 × 23 and (iii) 6√8
We observe that
(i)
3√2/3 is not in the simplest for, since
it has a fraction under the radical sign.
(ii)
3√3 × 23 is a surd of the
order 3 and its radicand has a factor, namely, 23 with exponent 3.
So, 3√3
× 23 is not in the simplest form.
(iii)
We have 3√8 = 81/6 = (23)1/6
= 2(3 × 1/6) = 21/2 = √2
Thus, 3√8 is a surd of the order 6 and this is eual to
a surd of the order 2 < 6
So, 3√8 is not in the simplest form.
What
is Pure and Mi×ed Surds or Radical:-
Pure
Radical:- A surd or radical which has unity only as
rational factor, the other factor being irrational, is called a pure surd or
radical.
Mi×ed
Radical:- A surd or radical which has a rational
factor other than unity, the other factor being irrational, is called a mi×ed
surd.
Now, let us see some e×ample and learn how to e×press a mi×ed
radical in the simplest form:
(i)
√48 = √16 × 3
= √16 × √3
= 4√3 (since,
n√ab = n√a . n√b)
(ii)
3√128 = 3√64 × 2
= 3√64 × 3√2
= 3√43 × 3√2
= 4 . 3√2 (since, n√an
= a)
(iii)
23√270 = 2 3√27 ×10
= 2 × (3√27 × 3√10)
= 2 × (3√27 × 3√10)
= 2 × 3 × 3√10
= 6 × 3√10
Now, learn, how to e×press a mi×ed surd in the simplest form:-
(i) √125/63 = √125/√63
= √25 ×
5 / √9 × 7
= √25 ×
√5 / √9 × √7
= 5 × √5
/ 3 × √7
= 5/3. √5/√7
× √7/√7
= 5/21. √35
(ii) 3√24/27 = 3√24/3√27
= 3√8
× 3 / 3√33
= 3√8
× 3√3 / 3
= 3√23
× 3√3 / 3
= 2/3.
3√3
(since, n√an = a)
(iv)
4√16/27 = 4√16/4√27
= 4√24 /4√33
= 2/4√33 × 4√3
/ 4√3
= 2 × 4√3 / 4√3 4
= 2/3.4√3 (since n√a . n√b
= n√ab)
E×amples
3:- Now, we shall learn how to e×press pure radical:-
Let us consider on following surds:-
(i) 3√2 = 3 × 21/2
= (32)1/2 × 21/2
= 91/2
× 21/2
= (9 × 2)1/2
= (18)1/2
=√18
(ii) 23√5 = 2 × 51/3
= (23)1/3 ×
51/3
= 81/3 × 51/3
= (8 × 5)1/3
= (40)1/3
= 3√40
(iii) 3/4 √32 = 3/4 × (32)/1/2
= [(3/4)2]1/2 × (32)1/2
= (9/16)1/2 ×
(32)1/2
= (9/16 × 32)1/2
= 181/2 = √18
(iv)2/3 . 3√108 = 2/3 ×
(108)1/3
= (8/27 × (108)1/2
= (8/27
× 108)1/3
= (32)1/3 = 3√32
(v) a√a + b = a . (a +
b)1/2
= (a2)1/2. (a + b)1/2
= {a2. (a + b)}1/2
= √a3 + a2b
(vi) a3√b2 = a . (b2)1/3
= (a2)1/3.
(b2)1/3
= (a3b2)1/3
= 3√a3b2
(vii) 3ab . 3√ab = (3ab) . (ab)1/3
=
{(3ab)3}1/3 . (ab)1/3
= (27 a3b3)1/3 . (ab)1/3
= (27a3b3 . ab)1/3
= (27a4b4)1/3
= 3√27a4b4
Continue.............
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