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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 a ^{n} 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 exponen**t of the power a^{n}.

**Laws of Indices:**

- where and (m, n)
- (ab)
^{n }= a^{n}b^{n} - 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) i**s called “pure surd”. e.g - A surd which has a rational factor other than unity, the other irrational, is called “
**mixed surd”**. e.g

**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:

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 :**

therefor y=3

** **