Few useful tricks to solve Surds questions for Competitive exams

A surd is said to be in its simplest form if the number under the root sign has no perfect square as a factor. For eg. √12 = √4 x √3 = 2√3. There are few questions on which questions are asked in the exam. There are direct trick and also very useful for time bound exams.

Note: n√a is a surd of nth term and it is irrational, whereas 'a' is a rational number

Law of Surds:

(a) n√a = a1/n 
(b) n√ab = n√a x n√b
(c) n√(a/b) = n√a/n√b 
(d) (n√a)n = a
(e) m√n√a = mn√a
(f) (n√a)m = n√am


How it derived: 

- Take Type(I): Here, the equation is √6+√6+√6+√6+....infinity
Hence, put this equation = x
Step-1:
                   √6+√6+√6+... infinity = x
                             Squaring both sides
                      6+√6+√6+√6+....infinity = x2
Step-2:
                           Now, using step 1,
                          6+x = x2    => x2-x-6 = 0           {solve this quadratic equation} 
                               you will get..... (x-2)(x-3)
for Answer:
- you get, x= √6+√6+√6+.... infinity = 2,3
to choose the correct answer, see the sign used. "+" sign has been used. Hence, answer will be 3.

If "-" sign would be there, √6-√6-√6-....infinity, answer will be 2 {as in the case of type (II) question}