Power, Indices and Surds Tips and Tricks
In this article we will discuss about Surds & Indices. This topic are very useful and acts as the base of simplification and Algebra. It is compulsory that question will comes 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:
where and (m, n)
- (ab)n = anbn
- 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 
is irrational, then 
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
- A surd which has a rational factor other than unity, the other irrational, is called “mixed surd”. e.g
- If is a surd it implies , a is a rational number and is an irrational number.
Quadratic Surd:
A surd of order 2 (i.e 
) is called a quadratic surd.
E.g. :
is a quadratic surd but 
is not a quadratic surd because 
is a rational number. Therefore 
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 :
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 
or 
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)
and 
then
e.g.
we have 
(2)
and then y=n
e.g. 
we have
therefor y=3
(3) If 
(4)If 
