Rationalizing Factor: When the product of two surds is a rational number, then each surd is called Rationalizing Factor (R F ) • Law of Surds and Exponents If a >
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A surd is an irrational number. In general, if x is rational, n is positive integer and if n x is irrational, then n x is called a surd of n th power. Here x is called radicand, n x is called radical sign and the index n" is called order of the surd. n x is read as n th root of x and can be written as 1/n xn x are called simple surds.
If it is a surd of n
th order, then (i) Whe = 2, it is called quadratic surd. (ii) Whe = 3, it is called cubic surd.Whe = 4, it is called biquadratic surd.
Note: -
Every surd is an irrational number but every irrational number is not a surd. So the representation of monomial surd on a number line is same that of irrational numbers. e.g., (i) is a surd and e is irrational number (ii) is an irrational number but it is not a surd (i) Pure Surd: A surd which has unity only as rational factor the other factor being irratio nal is called Pure Surd. e.g. 2,3443, 4, 5
(ii) Mixed Surd: A surd consisting of the product of a rational and irrational is called Mixed Surd e.g., 53, 12 ,and if a is rational number and not equal to zero and n b is a surd, then a n b, are mixed surd. If a = 1 they are called pure surd. Mixed Surd can be written as Pure Surd. www.plancess.com (iii) Compound Surd: A surd which is the sum or di?erence of two or more surds is called Compound Surd. e.g., 2 + 33, 1+2- 3
(iv) Monomial Surd: A surd consisting only one surd is called Monomial Surd. e.g., 3 5, 57 (v) Binomial Surd: A compound surd consisting of two surds is called a Binomial Surd. e.g.2 3 3, 3 7
(vi) Trinomial Surd: A compound surd consist of 3 surds is called Trinomial Surd. e.g.7 5 3 ,3 5 4 2 2 11
(vii) Similar Surds: If two surds are di?erent multiples of the same surd. ?ey are called Similar Surds otherwise they are Dissimilar Surds. e.g.,22,52 are similar surds and 33, 65 are dissimilar surds
(viii) Conjugate Surd: Two conjugate surds which are di?er only in signs (+/-) between them e.g., a + b and a - b are Conjugate Surds. Sometimes conjugate and reciprocal are same e.g., 2 -3 is conjugate of 2 +3 and reciprocal of 2 - 3 is 2 + 3
?e process of converting a surd to a rational number by multiplying it with a suitable RationalisingFactor.
When the product of two surds is a rational number, then each surd is called Rationalizing Factor (R.F.) e.g., ( 3 2)( 3- 2)= 3 - 2 = 1 which is rationalOne of R.F. of a
1/n is 11-n a e.g., 5 3/5 and 5 2/5 are Rationalising Factor of each otherR.F. of (a +b) is (a - b) and that of a- b is ab.
R.F. of
a b-c is ab c www.plancess.com (i) If a + b= c + d where a, c are rational number andb, dare surds, then a = c and b = d (ii) If a, b, 2 ab are positive rational numbers and b is a surd, then 22a a -ba- a -bab 22
(iii) If a, b, c, d are positive rational numbers and b, c, d are surds then bd bc cdab c d4 4 44c 4d 4b (iv) 2
4b k5a kb-b b5
(v) b-c3a bcc3 (vi) 33ab is a R.F. of a 2/3 - a 1/3 b 1/3 + b 2/3 and vice versa (vii) 33
ab is R.F. of a 2/3 + a 1/3 b 1/3 + b 2/3 and vice versa (viii) 22
x -kx -k (a b) (a- b) = 2a, a 2 - b = 1 x k1 If a > 0, b > 0 and n is a positive rational number then (i) n1nnn aaa (ii) nn n a b ab[Here order should be same] (iii) nnn aabb (iv) nmm nm n aa a (v) nppn aa np p/n aa or, npn mpm aa pnmp (a ) [Important for changing order of surds] (vi) 111mn
mn3n mnmnmmn a aa a a a (vii)
11 1 nmmmmnnmm n mn1n
n aa aa a a a www.plancess.com (viii) If a n = b then 1 nn ab a b (ix) m nmn aa If two surds are of same order then one whose radicand is larger is the larger of the two or if x > y > 0 and n > 1 is + ve integer then n x > n y e.g., 337719 13, 18 93
(i) 425 is a surd as radicand is a rational number.
Similar examples
5345, 12, 7 , 12, ....
(ii) 31 is a surd (as surd + rational number will give a surd)Similar examples
33 2 , 2 3 , 3 1, .....
(iii) 9 45is a surd as 9 45 is a perfect square of 25.Similar examples
7 4 3,7 4 3, 9 4 5,....
(iv) 13325 is a surd as
1 113181/2 1/336 18
((5) ) 5 5 5Similar examples
3453, 6,.....
(v) ?ese are not a surd: (a)93 is not a surd.
(b)1 5,because 15 is not a perfect square.
(c) 33 2, because radicand is an irrational number.
Illustration 1: If
1/3 2/3
x 33 3 ,then nd the value of 32x 9x 18x 12. Sol:
1/3 2/3
x 33 31/3 2/3
x33 3Cubing both sides
33 1/3 2/3
(x 3) 3 3 32x 9x 27x 27 12 3(3)(x 3)
1/3 2/3
since 3 3 x 3 32x 9x 18x 12 0 www.plancess.com
Illustration 2: If
xy a m,a n and2 y xz
a (m .n ) then ?nd the value of xyz.Sol: Given
2 y xz
a (m .n )2 xy yxz
a [(a ) .(a ) ] xy [ m a ,n a ]2 xy xy z
a [a .a ]2 2xy z
a [a ]2 2xyz
aaHere base is same
Hence 2 = 2xyz xyz = 1.
Illustration 3: Simplify
3 3 4 2.Sol: LCM of 3 and 4 is 12
121/3 4/12 4
33 3 And
121/4 3/12 3
22 234
32
12124 3 4312
3 2 (3 )(2 )
12 81 812 648