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.

Also Read : Concepts and Rules of Coordinate Geometry

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) mn√a = mn√a
(f) (n√a)m = n√am

Few point to remember:

– See the above types of surds mentioned, here i have solved the question of different types. These types of questions can be asked in the exam and then, you no need to solve the whole question. You can directly apply the formulas mentioned in the above image and get the answer.

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)
– 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}

All the types of questions are solved on this method and derived the formula, so that your precious time in the exams can be saved.