Indices

Base and Index:

If a real number is multiplied times in succession (where is a positive integer) then the product so obtained is called the -th power of and is written as (read as, to the power ). Thus, ….to   factors. Here, is called the base of and is called the index or exponent of . For example, . Here, is the base and 4 is the exponent of .

Note: In particular, is called the square of and is called the cube of

Root:

If and are two real numbers and is a positive integer such that , then is called the -th root of and is written as or .

Hence, it can be clearly seen that the -th root of () is such a number which when multiplied multiplied by itself times, i.e. it is such a number whose -th power is equal to .

In particular, if , then is called the second root or square root of and is written as or or simply, .

If , then is called the third root or cube root of and it is written as or .

For example,

Square root of 25 is 5, i.e.

[]

Cube root of 27 is 3, i.e.

[]

Sixth root of 64 is 2, i.e.

[]

Note:

1.

Again,

So, it is very clear that 5 and (-5) are both square roots of 25. As a result, when we try to find the square root of a positive number , we actually mean . Similarly, when we try to find the cube root of a positive number , there are 3 roots of out of which only one is positive. In general, we have roots when we try to find the -th root of a positive number of which only one root is positive.

For simplicity, when we want to find the square root or cube root or -th root of a real positive number we shall always mean only the positive real root. So

2. If is a real negative number and:

(i) is an odd positive integer then there exists no psitive -th root of but we shall always get a  real negative -th root of , say , such that . For example, if and , then or,

(ii) is an even positive integer then there exists no real number such that , i.e. in this case there is no real -th root of . For example has no real value, say , such that

Laws of Indices:

If a, b are two non-zero real numbers and m, n are positive integers then

(i)

This law is known as Fundamental Law of Index.

Proof: 

Since and are positive integers hence by definition we have,

and

[by definition]

(ii)

Proof:

Since and are positive integers, if $ m>n $, then is also a positive integer.

Hence by the positive law of index we have,

[ and are both positive integers]

or,

or,

Again,   and are positive integers when . Hence, the law can be similarly proved when

(iii)

Proof:

By definition, we have,

[by fundamental law]

(iv)

Proof:

(v)

Proof:

[using (iv)]

[ ]

or,

Exercise:

Evaluate:

a) 

b)

c)

d)

e)