Quant Notes on Power, Indices and Surds

By Parnab Mallick|Updated : February 12th, 2021

In this article, we will discuss Surds & Indices. This topic is very useful and acts as the base for simplification and Algebra. It is definite that question will come from this topic.

Indices and Surds

Let n be a positive integer and a be a real number, then:


where an is called "nth power of a" or "a raised to the power n"

where, a is called the base and n is called index or exponent of the power an.

Laws of Indices:

  • am × an = am+n where  and (m, n)
  • image006
  • image007
  • image008
  • image009
  • (ab)= anbn
  • image011
  • image012
  • image013image014
  • ap/q  =( a1/q )p  where p is a positive integer and q≠0
  • If the index of a power is unit (i.e. 1) then the value of the power is equal to its base, i.e.


Surds:If a is rational and n is a positive integer and image019 is irrational, then image020 is called a surds of order n or nth root of a.

  •  A surd which has unity as its rational factor (i.e., a = 1) is called "pure surd". e.g image023
  • A surd which has a rational factor other than unity, the other irrational, is called "mixed surd". e.gimage024


  • If image020is a surd it implies , a is a rational number  and image020is an irrational number.

Quadratic Surd:

A surd of order 2 (i.e image027) is called a quadratic surd.

E.g. :image028 is a quadratic surd but image029 is not a quadratic surd because image030 is a rational number. Therefore image031  is not a surd.


Cubic Surd:

A surd of order 3 is called a cubic surd. e.g.  91/3 

Important Formulae Based on Surds :

  • image032
  • image033
  • image034
  • image035
  • image036
  • image036
  • image037
  • image038
  • image039
  • image040
  • image041
  • image042
  • image043

Similar or like Surds:

surds having same irrational factors are called similar or like surds.

e.g.3√3,  4√3, 7√3 are similar surds.

Unlike surds:

Surds having no common irrational factors are called unlike surds.

e.g. 3√3, 7√5 are unlike surds.

Comparison of Surds:

If two surds are of the same order  then the one whose radicand is larger is the larger surds.

7√3  > 3√3.

If two surds are of different order then:

Question: Which is larger image044 or image045

Sol. Given surds are of order 2 & 3 respectively whose L.C.M. is 6.

Convert each into a surd of order 6, as shown below :




Some Useful Results :

(1) image049

and image050 thenimage051



we have image053



and image050then y=n

e.g. image057

we have image053

therefor y=3

(3) If image063



(4)If image065


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