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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 “n^th 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
• 
• 
• 
• a^p/q  =( a^1/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.  9^1/3 

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