Some of the important rules of surds are listed below.
1. Every rational number is not a surd.
2. Every irrational number is a surd.
3. A root of a positive real quantity is called a surd if its value cannot he exactly determined.
√9, ∛64, ∜(16/81) etc. are rational numbers but not surds because √9 = 3, ∛64 = 4, ∜(16/81) = 2/3 etc.
4. √a × √a = a ⇒ √5 × √5 = 5
5. The sum and difference of two simple quadratic surds are said to be conjugate surds or complementary surds to each other. Thus, (4√7 + √6) and (4√7 - √6) are surds conjugate to each other.
6. To express in the simplest form, denominator must be
7. The method of convening a given surd into a rational number on multiplication by another suitable surd is called rationalization of surds. In this case the multiplying surd is called the rationalizing factor of the given surd and conversely.
8. If a and b are both rationals and √x and √y are both surds and a + √x = b + √y then a = b and x = y
9. If a - √x = b - √y then a = b and x = y.
10. If a + √x = 0, then a = 0 and x = 0.
11. If a - √x = 0, then a = 0 and x = 0.`