Rules of Surds
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 rationalized.
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
When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.
Have a look at some more examples:
| Number | Simplified | As a Decimal | Surd or not? |
|---|---|---|---|
| √2 | √2 | 1.4142135...(etc) | Surd |
| √3 | √3 | 1.7320508...(etc) | Surd |
| √4 | 2 | 2 | Not a surd |
| √¼ | ½ | 0.5 | Not a surd |
| 3√11 | 3√11 | 2.2239800...(etc) | Surd |
| 3√27 | 3 | 3 | Not a surd |
| 5√3 | 5√3 | 1.2457309...(etc) | Surd |
The surds have a decimal which goes on forever without repeating, and are Irrational Numbers
 | In fact "Surd" used to be another name for "Irrational", but it is now used for a root that is irrational |