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High School Mathematics
2.12 Important Points in Radicals

1. Surd or Radical:
If 'n' is a natural number and 'a' is a positive rational number such that n a = a1/n is not a rational number, then na is called a surd of nth order.

2. Simple surd:
A surd which consists of a single term is called a simple or monomial surd.
3, 5, 6.

3. Mixed surd:
If 'a' is a rational number and b is a surd then a+b and a-b are called mixed surds.

4.Compound Surd:
A surd which is the sum or difference of two or more surds called a compound surd.
6 + 5, 3 - 53.

5. Binomial Surd:
A compound surd consisting of two surds is called a binomial surd.
Example: 6 + 5, 7 - 55

6. Similar Surd:
If two surds are different multiples of the same surd, they are called similar surds, otherwise they are dissimilar surds.
32, 52, 62 are similar surds.

7. Dissimilar Surds:
Surds which are not similar are called dissimilar surds.

8. Order of the surd:
In the surd na n is called the order of the surd.

9. Laws of Radicals:
1. na * nb = nab.
2. na / nb = n(a/b).
3. mna = m*na = nma.

10. Comparison of surds is possible only when they are of the same order, then the radicals are to be compared.

11. When the surds to be multiplied or divided are not of the same order, they have to be necessarily brought to same order before the required operation is done.

12. If the product of two surds is national then each of the two surds is called the rationalizing factor of the other.

13. The rationalizing factor of a given surd is not unique.

14. It is always convenient to use the simplest of all rationalizing factors of a given surd.

15. Quadratic surds are the surds of second order.

16. Two binomial expressions containing surds differing only conjugate each other.

17. If the product of two mixed surds a + b and a - b, where a and b , is a rational number then a + b and a - b are called conjugate surds to each other.

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