The order of biquadratic surd is
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Order of a Surd
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The order of a surd indicates the index of root to be extracted.
In n
√
a
, n is called the order of the surd and a is called the radicand.
For example: The order of the surd 5
√
z
is 5.
(i) A surd with index of root 2 is called a second order surd or quadratic surd.
The surds which have the indices of root 2 are called as second order surds or quadratic surds. For example√2, √3, √5, √7, √x are the surds of order 2.
Example: √2, √5, √10, √a, √m, √x, √(x + 1) are second order surd or quadratic surd (since the indices of roots are 2).
(ii) A surd with index of root 3 is called a third order surd or cubic surd.
If x is a positive integer with nth root, then is a surd of nth order when the value of is irrational. In expression n is the order of surd and x is called as radicand. For example is surd of order 3.
The surds which have the indices of cube roots are called as third order surds or cubic surds. For example ∛2, ∛3, ∛10, ∛17, ∛x are the surds of order 3 or cubic surds.
Example: ∛2, ∛5, ∛7, ∛15, ∛100, ∛a, ∛m, ∛x, ∛(x - 1) are third order surd or cubic surd (since the indices of roots are 3).
(iii) A surd with index of root 4 is called a fourth order surd.
The surds which have the indices of four roots are called as forth order surds or bi-quadratic surds.
For example ∜2, ∜4, ∜9, ∜20, ∜x are the surds of order 4.
Example: 4
√
2
, 4
√
3
, 4
√
9
, 4
√
17
, 4
√
70
, 4
√
a
, 4
√
m
, 4
√
x
, 4
√
x−1
are third order surd or cubic surd (since the indices of roots are 4).
(iv) In general, a surd with index of root n is called a nth order surd.
Similarly the surds which have the indices of n roots are nth order surds. n
√
2
, n
√
17
, n
√
19
, n
√
x
are the surds of order n.
Example: n
√
2
, n
√
3
, n
√
9
, n
√
17
, n
√
70
, n
√
a
, n
√
m
, n
√
x
, n
√
x−1
are nth order surd (since the indices of roots are n).
Problem on finding the order of a surd:
Express ∛4 as a surd of order 12.
Solution:
Now, ∛4
= 41/3
= 4
1×4
3×4
, [Since, we are to convert order 3 into 12, so we multiply both numerator and denominator of 1/3 by 4]
= 44/12
= 12
√
44
= 12
√
256
Problems on finding the order of surds:
1. Express √2 as a surd of order 6.
Solution:
√2 = 21/2
= 2
1×3
2×3
= 2
3
6
= 81/6
= 6
√
8
So 6
√
8
is a surd of order 6.
2. Express ∛3 as a surd of order 9.
Solution:
∛3 = 31/3
= 3
1×3
3×3
= 3
3
9
= 271/9
= 9
√
27
So 9
√
27
is a surd of order 9.
3. Simplify the surd ∜25 to a quadratic surd.
Solution:
∜25 = 251/4
= 5
2×1
4
= 3
1
2
= 2
√
5
= √5
So √5 is a surd of order 2 or a quadratic surd.
11 and 12 Grade Math
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